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Theorem sbcoreleleqVD 39679
Description: Virtual deduction proof of sbcoreleleq 39345. 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 39446 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 𝑦𝑥𝐴)   )
3:1,?: e1a 39446 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑦 𝑥𝐴𝑥)   )
4:1,?: e1a 39446 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
5:2,3,4,?: e111 39493 (   𝐴𝐵   ▶   ((𝑥𝐴 𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥 [𝐴 / 𝑦]𝑥 = 𝑦))   )
6:1,?: e1a 39446 (   𝐴𝐵    ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
7:5,6: e11 39507 (   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 3640 . . . . . . 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 1108 . . . . . . . . 9 ((𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦))
43bicomi 215 . . . . . . . 8 (((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
54sbcbii 3652 . . . . . . 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 3640 . . . . . . . 8 ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥))
87a1i 11 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥)))
98orbi1d 940 . . . . . 6 (𝐴𝐵 → (([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
102, 6, 93bitr3d 300 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
11 df-3or 1108 . . . . 5 (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦))
1210, 11syl6bbr 280 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))
1312dfvd1ir 39383 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
14 sbcel2gv 3656 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴))
1514dfvd1ir 39383 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴)   )
16 sbcel1v 3655 . . . 4 ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥)
17 eqsbc3r 3653 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴))
1817dfvd1ir 39383 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
19 3orbi123 39321 . . . . 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 39290 . . . 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 39497 . . 3 (   𝐴𝐵   ▶   ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
22 biantr 840 . . 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 39508 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴))   )
2423in1 39381 1 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wo 873  w3o 1106   = wceq 1652  wcel 2155  [wsbc 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352  df-sbc 3597  df-vd1 39380
This theorem is referenced by: (None)
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