Theorem sbcie2s 17129

Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.)

Hypotheses

Ref	Expression
sbcie2s.a	⊢ 𝐴 = (𝐸‘𝑊)
sbcie2s.b	⊢ 𝐵 = (𝐹‘𝑊)
sbcie2s.1	⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcie2s	⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑎,𝑏,𝑤   𝐸,𝑎,𝑏   𝐹,𝑏   𝑊,𝑎,𝑏   𝜓,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏)   𝜓(𝑤)   𝐴(𝑤,𝑎,𝑏)   𝐵(𝑤,𝑎,𝑏)   𝐸(𝑤)   𝐹(𝑤,𝑎)   𝑊(𝑤)

Proof of Theorem sbcie2s

Step	Hyp	Ref	Expression
1		fvex 6847	. 2 ⊢ (𝐸‘𝑤) ∈ V
2		fvex 6847	. 2 ⊢ (𝐹‘𝑤) ∈ V
3		fveq2 6834	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
4		sbcie2s.a	. . . . . 6 ⊢ 𝐴 = (𝐸‘𝑊)
5	3, 4	eqtr4di 2793	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴)
6	5	eqeq2d 2751	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) ↔ 𝑎 = 𝐴))
7	6	biimpd 230	. . 3 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) → 𝑎 = 𝐴))
8		fveq2 6834	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
9		sbcie2s.b	. . . . . 6 ⊢ 𝐵 = (𝐹‘𝑊)
10	8, 9	eqtr4di 2793	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵)
11	10	eqeq2d 2751	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑏 = (𝐹‘𝑤) ↔ 𝑏 = 𝐵))
12	11	biimpd 230	. . 3 ⊢ (𝑤 = 𝑊 → (𝑏 = (𝐹‘𝑤) → 𝑏 = 𝐵))
13		sbcie2s.1	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))
14	13	a1i 11	. . 3 ⊢ (𝑤 = 𝑊 → ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)))
15	7, 12, 14	syl2and 614	. 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜑 ↔ 𝜓)))
16	1, 2, 15	sbc2iedv 3806	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  [wsbc 3730  ‘cfv 6492

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500

This theorem is referenced by:  isassa  21838  istrkgc  28547  istrkgb  28548  istrkge  28550  istrkgl  28551  ishpg  28852  iscgra  28902