Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcoreleleq Structured version   Visualization version   GIF version

Theorem sbcoreleleq 39521
Description: Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 39855. (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 sbcel2gv 3693 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝑦𝑥𝐴))
2 sbcel1v 3692 . . . 4 ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥)
32a1i 11 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥))
4 eqsbc3r 3690 . . 3 (𝐴𝑉 → ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴))
5 3orbi123 39497 . . . 4 ((([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) ∧ ([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) ∧ ([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴)) → (([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
653impexpbicomi 39466 . . 3 (([𝐴 / 𝑦]𝑥𝑦𝑥𝐴) → (([𝐴 / 𝑦]𝑦𝑥𝐴𝑥) → (([𝐴 / 𝑦]𝑥 = 𝑦𝑥 = 𝐴) → ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))))
71, 3, 4, 6syl3c 66 . 2 (𝐴𝑉 → ((𝑥𝐴𝐴𝑥𝑥 = 𝐴) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦)))
8 sbc3or 39518 . 2 ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ ([𝐴 / 𝑦]𝑥𝑦[𝐴 / 𝑦]𝑦𝑥[𝐴 / 𝑦]𝑥 = 𝑦))
97, 8syl6rbbr 282 1 (𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3o 1107   = wceq 1653  wcel 2157  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634
This theorem is referenced by:  tratrb  39522  tratrbVD  39857
  Copyright terms: Public domain W3C validator