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 = (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 6241	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		simpr 476	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤))
3		fveq2 6229	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
4		sbcies.a	. . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊)
5	3, 4	syl6reqr 2704	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤))
6	5	adantr 480	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤))
7	2, 6	eqtr4d 2688	. . . 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 213	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑))
11	1, 10	sbcied 3505	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))

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)