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Theorem sbcoreleleqVD 43610
Description: Virtual deduction proof of sbcoreleleq 43286. 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 43378 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 𝑦𝑥𝐴)   )
3:1,?: e1a 43378 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑦 𝑥𝐴𝑥)   )
4:1,?: e1a 43378 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
5:2,3,4,?: e111 43425 (   𝐴𝐵   ▶   ((𝑥𝐴 𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥 [𝐴 / 𝑦]𝑥 = 𝑦))   )
6:1,?: e1a 43378 (   𝐴𝐵    ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
7:5,6: e11 43439 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥 𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴))   )
qed:7: (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦 𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcoreleleqVD (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem sbcoreleleqVD
StepHypRef Expression
1 sbcor 3830 . . . . . . 7 ([𝐴 / 𝑦]((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦))
21a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑦]((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
3 df-3or 1088 . . . . . . . . 9 ((𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦))
43bicomi 223 . . . . . . . 8 (((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
54sbcbii 3837 . . . . . . 7 ([𝐴 / 𝑦]((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ [𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦))
65a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑦]((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ [𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
7 sbcor 3830 . . . . . . . 8 ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥))
87a1i 11 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥)))
98orbi1d 915 . . . . . 6 (𝐴𝐵 → (([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
102, 6, 93bitr3d 308 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
11 df-3or 1088 . . . . 5 (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦))
1210, 11bitr4di 288 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))
1312dfvd1ir 43324 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
14 sbcel2gv 3849 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴))
1514dfvd1ir 43324 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴)   )
16 sbcel1v 3848 . . . 4 ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥)
17 eqsbc2 3846 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴))
1817dfvd1ir 43324 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
19 3orbi123 43262 . . . . 5 ((([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) ∧ ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) ∧ ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)) → (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
20193impexpbicomi 43231 . . . 4 (([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) → (([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) → (([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴) → ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))))
2115, 16, 18, 20e101 43429 . . 3 (   𝐴𝐵   ▶   ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
22 biantr 804 . . 3 ((([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)) ∧ ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))) → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
2313, 21, 22e11an 43440 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴))   )
2423in1 43322 1 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845  w3o 1086   = wceq 1541  wcel 2106  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sbc 3778  df-vd1 43321
This theorem is referenced by: (None)
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