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 = (𝐸‘𝑊)
sbcies.1	⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcies	⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))

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

Step	Hyp	Ref	Expression
1		fvexd 6689	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		simpr 488	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤))
3		sbcies.a	. . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊)
4		fveq2 6674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5	3, 4	eqtr4id 2792	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤))
6	5	adantr 484	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤))
7	2, 6	eqtr4d 2776	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = 𝐴)
8		sbcies.1	. . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))
9	7, 8	syl 17	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜑 ↔ 𝜓))
10	9	bicomd 226	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑))
11	1, 10	sbcied 3724	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))

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)