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Theorem sbc3orgVD 43225
Description: Virtual deduction proof of the analogue of sbcor 3796 with three disjuncts. 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 43001 (   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 43068 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑 𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
4:1,33,?: e11 43062 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
5:2,4,?: e11 43062 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
6:1,?: e1a 43001 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
7:6,?: e1a 43001 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) [𝐴 / 𝑥]𝜒))   )
8:5,7,?: e11 43062 (   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 43068 (   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 42948 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 sbcor 3796 . . . . . . 7 ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))
32a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒)))
41, 3e1a 43001 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
5 df-3or 1089 . . . . . . . . 9 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
65bicomi 223 . . . . . . . 8 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
76ax-gen 1798 . . . . . . 7 𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))
8 spsbc 3756 . . . . . . 7 (𝐴𝐵 → (∀𝑥(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) → [𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))))
91, 7, 8e10 43068 . . . . . 6 (   𝐴𝐵   ▶   [𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒))   )
10 sbcbig 3797 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) ↔ ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))))
1110biimpd 228 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](((𝜑𝜓) ∨ 𝜒) ↔ (𝜑𝜓𝜒)) → ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))))
121, 9, 11e11 43062 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]((𝜑𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑𝜓𝜒))   )
13 bitr3 353 . . . . . 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))))
1413com12 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))))
154, 12, 14e11 43062 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
16 sbcor 3796 . . . . . . 7 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
1716a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
181, 17e1a 43001 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
19 orbi1 917 . . . . 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)))
2018, 19e1a 43001 . . . 4 (   𝐴𝐵   ▶   (([𝐴 / 𝑥](𝜑𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
21 bibi1 352 . . . . 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))))
2221biimprd 248 . . . 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))))
2315, 20, 22e11 43062 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   )
24 df-3or 1089 . . . 4 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))
2524bicomi 223 . . 3 ((([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
26 bibi1 352 . . . 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))))
2726biimprd 248 . . 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))))
2823, 25, 27e10 43068 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
2928in1 42945 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 205  wo 846  w3o 1087  wal 1540  wcel 2107  [wsbc 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-sbc 3744  df-vd1 42944
This theorem is referenced by: (None)
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