Theorem sbcies 28706

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		fvex 6113	. . 3 ⊢ (𝐸‘𝑤) ∈ V
2	1	a1i 11	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
3		simpr 476	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤))
4		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5		sbcies.a	. . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊)
6	4, 5	syl6reqr 2663	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤))
7	6	adantr 480	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤))
8	3, 7	eqtr4d 2647	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = 𝐴)
9		sbcies.1	. . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))
10	8, 9	syl 17	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜑 ↔ 𝜓))
11	10	bicomd 212	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑))
12	2, 11	sbcied 3439	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by: (None)