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Theorem sbcoreleleqVD 41348
Description: Virtual deduction proof of sbcoreleleq 41024. 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:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1,?: e1a 41116 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)   ) 3:1,?: e1a 41116 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)   ) 4:1,?: e1a 41116 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝐴)   ) 5:2,3,4,?: e111 41163 ⊢ (   𝐴 ∈ 𝐵   ▶   ((𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   ) 6:1,?: e1a 41116 ⊢ (   𝐴 ∈ 𝐵    ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝑦 ∨ [𝐴 / 𝑦]𝑦 ∈ 𝑥 ∨ [𝐴 / 𝑦]𝑥 = 𝑦))   ) 7:5,6: e11 41177 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 𝐴))   ) qed:7: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 ∨ 𝐴 ∈ 𝑥 ∨ 𝑥 = 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 3798 . . . . . . 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 1085 . . . . . . . . 9 ((𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦))
43bicomi 227 . . . . . . . 8 (((𝑥𝑦𝑦𝑥) ∨ 𝑥 = 𝑦) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
54sbcbii 3805 . . . . . . 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 3798 . . . . . . . 8 ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥))
87a1i 11 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥)))
98orbi1d 914 . . . . . 6 (𝐴𝐵 → (([𝐴 / 𝑦](𝑥𝑦𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
102, 6, 93bitr3d 312 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦)))
11 df-3or 1085 . . . . 5 (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥) ∨ [𝐴 / 𝑦]𝑥 = 𝑦))
1210, 11syl6bbr 292 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))
1312dfvd1ir 41062 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
14 sbcel2gv 3816 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴))
1514dfvd1ir 41062 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴)   )
16 sbcel1v 3815 . . . 4 ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥)
17 eqsbc3r 3813 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴))
1817dfvd1ir 41062 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)   )
19 3orbi123 41000 . . . . 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 40969 . . . 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 41167 . . 3 (   𝐴𝐵   ▶   ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))   )
22 biantr 805 . . 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 41178 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴))   )
2423in1 41060 1 (𝐴𝐵 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2115  [wsbc 3749 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-sbc 3750  df-vd1 41059 This theorem is referenced by: (None)
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