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Theorem sbc3orgVD 38906
Description: Virtual deduction proof of sbc3orgOLD 38562. 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:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1,?: e1a 38672 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) [𝐴 / 𝑥]𝜒))   )
3:: (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 𝜓𝜒))
32:3: 𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
33:1,32,?: e10 38739 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑 𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
4:1,33,?: e11 38733 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
5:2,4,?: e11 38733 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
6:1,?: e1a 38672 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
7:6,?: e1a 38672 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) [𝐴 / 𝑥]𝜒))   )
8:5,7,?: e11 38733 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) [𝐴 / 𝑥]𝜒))   )
9:?: ((([𝐴 / 𝑥]𝜑 [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑 [𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
10:8,9,?: e10 38739 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒))   )
qed:10: (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbc3orgVD (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Proof of Theorem sbc3orgVD
StepHypRef Expression
1 idn1 38610 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 sbcorgOLD 38560 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)))
31, 2e1a 38672 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
4 df-3or 1037 . . . . . . . . 9 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
54bicomi 214 . . . . . . . 8 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
65ax-gen 1720 . . . . . . 7 𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
7 spsbc 3442 . . . . . . 7 (𝐴𝐵 → (∀𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) → [𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))))
81, 6, 7e10 38739 . . . . . 6 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
9 sbcbig 3474 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) ↔ ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))))
109biimpd 219 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))))
111, 8, 10e11 38733 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
12 bitr3 342 . . . . . 6 (([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒)) → (([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))))
1312com12 32 . . . . 5 (([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒)) → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))))
143, 11, 13e11 38733 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
15 sbcorgOLD 38560 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
161, 15e1a 38672 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
17 orbi1 741 . . . . 5 (([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)) → (([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)))
1816, 17e1a 38672 . . . 4 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
19 bibi1 341 . . . . 5 (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)) ↔ (([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))))
2019biimprd 238 . . . 4 (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)) → ((([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))))
2114, 18, 20e11 38733 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
22 df-3or 1037 . . . 4 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))
2322bicomi 214 . . 3 ((([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
24 bibi1 341 . . . 4 (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) ↔ ((([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
2524biimprd 238 . . 3 (([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒)) → (((([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
2621, 23, 25e10 38739 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
2726in1 38607 1 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  w3o 1035  wal 1479  wcel 1988  [wsbc 3429
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-9 1997  ax-10 2017  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430  df-vd1 38606
This theorem is referenced by: (None)
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