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Theorem bnj1109 30983
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1109.1 𝑥((𝐴𝐵𝜑) → 𝜓)
bnj1109.2 ((𝐴 = 𝐵𝜑) → 𝜓)
Assertion
Ref Expression
bnj1109 𝑥(𝜑𝜓)

Proof of Theorem bnj1109
StepHypRef Expression
1 bnj1109.2 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝜓)
21ex 449 . . . . . 6 (𝐴 = 𝐵 → (𝜑𝜓))
32a1i 11 . . . . 5 ((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
43ax-gen 1762 . . . 4 𝑥((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
5 bnj1109.1 . . . . 5 𝑥((𝐴𝐵𝜑) → 𝜓)
6 impexp 461 . . . . . 6 (((𝐴𝐵𝜑) → 𝜓) ↔ (𝐴𝐵 → (𝜑𝜓)))
76exbii 1814 . . . . 5 (∃𝑥((𝐴𝐵𝜑) → 𝜓) ↔ ∃𝑥(𝐴𝐵 → (𝜑𝜓)))
85, 7mpbi 220 . . . 4 𝑥(𝐴𝐵 → (𝜑𝜓))
9 exintr 1859 . . . 4 (∀𝑥((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓))) → (∃𝑥(𝐴𝐵 → (𝜑𝜓)) → ∃𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓)))))
104, 8, 9mp2 9 . . 3 𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓)))
11 exancom 1827 . . 3 (∃𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓))) ↔ ∃𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))))
1210, 11mpbi 220 . 2 𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓)))
13 df-ne 2824 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
1413imbi1i 338 . . 3 ((𝐴𝐵 → (𝜑𝜓)) ↔ (¬ 𝐴 = 𝐵 → (𝜑𝜓)))
15 pm2.61 183 . . . 4 ((𝐴 = 𝐵 → (𝜑𝜓)) → ((¬ 𝐴 = 𝐵 → (𝜑𝜓)) → (𝜑𝜓)))
1615imp 444 . . 3 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (¬ 𝐴 = 𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1714, 16sylan2b 491 . 2 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1812, 17bnj101 30917 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ne 2824
This theorem is referenced by:  bnj1030  31181  bnj1128  31184  bnj1145  31187
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