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Theorem sbcies 31983
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 6906 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 simpr 485 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = (𝐸𝑤))
3 sbcies.a . . . . . . 7 𝐴 = (𝐸𝑊)
4 fveq2 6891 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
53, 4eqtr4id 2791 . . . . . 6 (𝑤 = 𝑊𝐴 = (𝐸𝑤))
65adantr 481 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝐴 = (𝐸𝑤))
72, 6eqtr4d 2775 . . . 4 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = 𝐴)
8 sbcies.1 . . . 4 (𝑎 = 𝐴 → (𝜑𝜓))
97, 8syl 17 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜑𝜓))
109bicomd 222 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜓𝜑))
111, 10sbcied 3822 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  Vcvv 3474  [wsbc 3777  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551
This theorem is referenced by: (None)
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