Theorem sbcies 32467

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 6837	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		simpr 484	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝑎 = (𝐸‘𝑤))
3		sbcies.a	. . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊)
4		fveq2 6822	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5	3, 4	eqtr4id 2785	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝐴 = (𝐸‘𝑤))
6	5	adantr 480	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → 𝐴 = (𝐸‘𝑤))
7	2, 6	eqtr4d 2769	. . . 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 223	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝜓 ↔ 𝜑))
11	1, 10	sbcied 3780	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Vcvv 3436  [wsbc 3736  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by: (None)