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Theorem sbcoreleleq 42909
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 43233. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcoreleleq (𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcoreleleq
StepHypRef Expression
1 sbc3or 42906 . 2 ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))
2 sbcel2gv 3815 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴))
3 sbcel1v 3814 . . . 4 ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥)
43a1i 11 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥))
5 eqsbc2 3812 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴))
6 3orbi123 42885 . . . 4 ((([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) ∧ ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) ∧ ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)) → (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
763impexpbicomi 42854 . . 3 (([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) → (([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) → (([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴) → ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))))
82, 4, 5, 7syl3c 66 . 2 (𝐴𝑉 → ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))
91, 8bitr4id 290 1 (𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3o 1087   = wceq 1542  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-3an 1090  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
This theorem is referenced by:  tratrb  42910  tratrbVD  43235
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