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Theorem sbcies 30410
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 6689 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 simpr 488 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = (𝐸𝑤))
3 sbcies.a . . . . . . 7 𝐴 = (𝐸𝑊)
4 fveq2 6674 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
53, 4eqtr4id 2792 . . . . . 6 (𝑤 = 𝑊𝐴 = (𝐸𝑤))
65adantr 484 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝐴 = (𝐸𝑤))
72, 6eqtr4d 2776 . . . 4 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = 𝐴)
8 sbcies.1 . . . 4 (𝑎 = 𝐴 → (𝜑𝜓))
97, 8syl 17 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜑𝜓))
109bicomd 226 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜓𝜑))
111, 10sbcied 3724 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  Vcvv 3398  [wsbc 3680  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347
This theorem is referenced by: (None)
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