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Theorem sbcies 29450
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 6241 . 2 (𝑤 = 𝑊 → (𝐸𝑤) ∈ V)
2 simpr 476 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = (𝐸𝑤))
3 fveq2 6229 . . . . . . 7 (𝑤 = 𝑊 → (𝐸𝑤) = (𝐸𝑊))
4 sbcies.a . . . . . . 7 𝐴 = (𝐸𝑊)
53, 4syl6reqr 2704 . . . . . 6 (𝑤 = 𝑊𝐴 = (𝐸𝑤))
65adantr 480 . . . . 5 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝐴 = (𝐸𝑤))
72, 6eqtr4d 2688 . . . 4 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → 𝑎 = 𝐴)
8 sbcies.1 . . . 4 (𝑎 = 𝐴 → (𝜑𝜓))
97, 8syl 17 . . 3 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜑𝜓))
109bicomd 213 . 2 ((𝑤 = 𝑊𝑎 = (𝐸𝑤)) → (𝜓𝜑))
111, 10sbcied 3505 1 (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  Vcvv 3231  [wsbc 3468  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by: (None)
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