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Theorem tratrbVD 38917
Description: Virtual deduction proof of tratrb 38566. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ) 2:1,?: e1a 38672 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   ) 3:1,?: e1a 38672 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   ) 4:1,?: e1a 38672 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   ) 5:: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ) 6:5,?: e2 38676 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝑦   ) 7:5,?: e2 38676 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑦 ∈ 𝐵   ) 8:2,7,4,?: e121 38701 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑦 ∈ 𝐴   ) 9:2,6,8,?: e122 38698 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝐴   ) 10:: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥   ▶   𝐵 ∈ 𝑥   ) 11:6,7,10,?: e223 38680 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝐵 ∈ 𝑥   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)   ) 12:11: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥))   ) 13:: ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) 14:12,13,?: e20 38774 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   ¬ 𝐵 ∈ 𝑥   ) 15:: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   𝑥 = 𝐵   ) 16:7,15,?: e23 38802 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   𝑦 ∈ 𝑥   ) 17:6,16,?: e23 38802 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵), 𝑥 = 𝐵   ▶   (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)   ) 18:17: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))   ) 19:: ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) 20:18,19,?: e20 38774 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   ¬ 𝑥 = 𝐵   ) 21:3,?: e1a 38672 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   ) 22:21,9,4,?: e121 38701 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   ) 23:22,?: e2 38676 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   ) 24:4,23,?: e12 38771 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵)   ) 25:14,20,24,?: e222 38681 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴), (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)   ▶   𝑥 ∈ 𝐵   ) 26:25: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   ) 27:: ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) 28:27,?: e0a 38819 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)) 29:28,26: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)    ▶   ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   ) 30:: ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑥∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) 31:30,?: e0a 38819 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)) 32:31,29: ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)   ) 33:32,?: e1a 38672 ⊢ (   (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   ) qed:33: ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tratrbVD ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem tratrbVD
StepHypRef Expression
1 hbra1 2939 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
2 alrim3con13v 38563 . . . . 5 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑥𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
31, 2e0a 38819 . . . 4 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
4 ax-5 1837 . . . . . . 7 (𝑥𝐴 → ∀𝑦 𝑥𝐴)
5 hbra1 2939 . . . . . . 7 (∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
64, 5hbral 2940 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
7 alrim3con13v 38563 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)))
86, 7e0a 38819 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑦(Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴))
9 idn2 38658 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝑦𝑦𝐵)   )
10 simpl 473 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑥𝑦)
119, 10e2 38676 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝑦   )
12 simpr 477 . . . . . . . . . . 11 ((𝑥𝑦𝑦𝐵) → 𝑦𝐵)
139, 12e2 38676 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐵   )
14 idn3 38660 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   𝐵𝑥   )
15 pm3.2an3 1238 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝐵 → (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))))
1611, 13, 14, 15e223 38680 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝐵𝑥   ▶   (𝑥𝑦𝑦𝐵𝐵𝑥)   )
1716in3 38654 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥))   )
18 en3lp 8498 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝐵𝐵𝑥)
19 con3 149 . . . . . . . 8 ((𝐵𝑥 → (𝑥𝑦𝑦𝐵𝐵𝑥)) → (¬ (𝑥𝑦𝑦𝐵𝐵𝑥) → ¬ 𝐵𝑥))
2017, 18, 19e20 38774 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝐵𝑥   )
21 idn3 38660 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
22 eleq2 2688 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
2322biimprcd 240 . . . . . . . . . . 11 (𝑦𝐵 → (𝑥 = 𝐵𝑦𝑥))
2413, 21, 23e23 38802 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   𝑦𝑥   )
25 pm3.2 463 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑥 → (𝑥𝑦𝑦𝑥)))
2611, 24, 25e23 38802 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ,   𝑥 = 𝐵   ▶   (𝑥𝑦𝑦𝑥)   )
2726in3 38654 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥))   )
28 en2lp 8495 . . . . . . . 8 ¬ (𝑥𝑦𝑦𝑥)
29 con3 149 . . . . . . . 8 ((𝑥 = 𝐵 → (𝑥𝑦𝑦𝑥)) → (¬ (𝑥𝑦𝑦𝑥) → ¬ 𝑥 = 𝐵))
3027, 28, 29e20 38774 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶    ¬ 𝑥 = 𝐵   )
31 idn1 38610 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   )
32 simp3 1061 . . . . . . . . 9 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → 𝐵𝐴)
3331, 32e1a 38672 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝐵𝐴   )
34 simp2 1060 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3531, 34e1a 38672 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
36 ralcom2 3099 . . . . . . . . . . 11 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → ∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
3735, 36e1a 38672 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
38 simp1 1059 . . . . . . . . . . . 12 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐴)
3931, 38e1a 38672 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐴   )
40 trel 4750 . . . . . . . . . . . . 13 (Tr 𝐴 → ((𝑦𝐵𝐵𝐴) → 𝑦𝐴))
4140expd 452 . . . . . . . . . . . 12 (Tr 𝐴 → (𝑦𝐵 → (𝐵𝐴𝑦𝐴)))
4239, 13, 33, 41e121 38701 . . . . . . . . . . 11 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑦𝐴   )
43 trel 4750 . . . . . . . . . . . 12 (Tr 𝐴 → ((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
4443expd 452 . . . . . . . . . . 11 (Tr 𝐴 → (𝑥𝑦 → (𝑦𝐴𝑥𝐴)))
4539, 11, 42, 44e122 38698 . . . . . . . . . 10 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐴   )
46 rspsbc2 38564 . . . . . . . . . . 11 (𝐵𝐴 → (𝑥𝐴 → (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4746com13 88 . . . . . . . . . 10 (∀𝑦𝐴𝑥𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐴 → (𝐵𝐴[𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))))
4837, 45, 33, 47e121 38701 . . . . . . . . 9 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
49 equid 1937 . . . . . . . . . . 11 𝑥 = 𝑥
50 sbceq2a 3441 . . . . . . . . . . 11 (𝑥 = 𝑥 → ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
5149, 50ax-mp 5 . . . . . . . . . 10 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5251biimpi 206 . . . . . . . . 9 ([𝑥 / 𝑥][𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
5348, 52e2 38676 . . . . . . . 8 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   [𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
54 sbcoreleleq 38565 . . . . . . . . 9 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5554biimpd 219 . . . . . . . 8 (𝐵𝐴 → ([𝐵 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝑥𝐵𝐵𝑥𝑥 = 𝐵)))
5633, 53, 55e12 38771 . . . . . . 7 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   (𝑥𝐵𝐵𝑥𝑥 = 𝐵)   )
57 3ornot23 38535 . . . . . . . 8 ((¬ 𝐵𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵))
5857ex 450 . . . . . . 7 𝐵𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥𝐵𝐵𝑥𝑥 = 𝐵) → 𝑥𝐵)))
5920, 30, 56, 58e222 38681 . . . . . 6 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ,   (𝑥𝑦𝑦𝐵)   ▶   𝑥𝐵   )
6059in2 38650 . . . . 5 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   ((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
618, 60gen11nv 38662 . . . 4 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
623, 61gen11nv 38662 . . 3 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵)   )
63 dftr2 4745 . . . 4 (Tr 𝐵 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵))
6463biimpri 218 . . 3 (∀𝑥𝑦((𝑥𝑦𝑦𝐵) → 𝑥𝐵) → Tr 𝐵)
6562, 64e1a 38672 . 2 (   (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴)   ▶   Tr 𝐵   )
6665in1 38607 1 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1035   ∧ w3a 1036  ∀wal 1479   = wceq 1481   ∈ wcel 1988  ∀wral 2909  [wsbc 3429  Tr wtr 4743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934  ax-reg 8482 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-fr 5063  df-vd1 38606  df-vd2 38614  df-vd3 38626 This theorem is referenced by: (None)
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