Theorem sbcie2s 17100

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 6855	. 2 ⊢ (𝐸‘𝑤) ∈ V
2		fvex 6855	. 2 ⊢ (𝐹‘𝑤) ∈ V
3		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
4		sbcie2s.a	. . . . . 6 ⊢ 𝐴 = (𝐸‘𝑊)
5	3, 4	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴)
6	5	eqeq2d 2748	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) ↔ 𝑎 = 𝐴))
7	6	biimpd 229	. . 3 ⊢ (𝑤 = 𝑊 → (𝑎 = (𝐸‘𝑤) → 𝑎 = 𝐴))
8		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
9		sbcie2s.b	. . . . . 6 ⊢ 𝐵 = (𝐹‘𝑊)
10	8, 9	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵)
11	10	eqeq2d 2748	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑏 = (𝐹‘𝑤) ↔ 𝑏 = 𝐵))
12	11	biimpd 229	. . 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 609	. 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜑 ↔ 𝜓)))
16	1, 2, 15	sbc2iedv 3819	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  [wsbc 3742  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508

This theorem is referenced by:  isassa  21823  istrkgc  28538  istrkgb  28539  istrkge  28541  istrkgl  28542  ishpg  28843  iscgra  28893