Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcies Structured version   Visualization version   GIF version

Theorem sbcies 32508
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
sbcies.a 𝐴 = (𝐸𝑊)
sbcies.1 (𝑎 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbcies (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
Distinct variable groups:   𝑤,𝑎   𝐸,𝑎   𝑊,𝑎   𝜑,𝑎
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑤,𝑎)   𝐴(𝑤,𝑎)   𝐸(𝑤)   𝑊(𝑤)

Proof of Theorem sbcies
StepHypRef Expression
1 fvexd 6920 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 simpr 484 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = (𝐸𝑤))
3 sbcies.a . . . . . . 7 𝐴 = (𝐸𝑊)
4 fveq2 6905 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
53, 4eqtr4id 2795 . . . . . 6 (𝑤 = 𝑊𝐴 = (𝐸𝑤))
65adantr 480 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝐴 = (𝐸𝑤))
72, 6eqtr4d 2779 . . . 4 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = 𝐴)
8 sbcies.1 . . . 4 (𝑎 = 𝐴 → (𝜑𝜓))
97, 8syl 17 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜑𝜓))
109bicomd 223 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜓𝜑))
111, 10sbcied 3831 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  Vcvv 3479  [wsbc 3787  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator