Theorem sbcie2s 16907

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

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]𝜓 ↔ 𝜑))

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

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑤,𝑎,𝑏)   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 6817	. 2 ⊢ (𝐸‘𝑤) ∈ V
2		fvex 6817	. 2 ⊢ (𝐹‘𝑤) ∈ V
3		simprl 769	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = (𝐸‘𝑤))
4		fveq2 6804	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5		sbcie2s.a	. . . . . . . 8 ⊢ 𝐴 = (𝐸‘𝑊)
6	4, 5	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴)
7	6	adantr 482	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐸‘𝑤) = 𝐴)
8	3, 7	eqtrd 2776	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = 𝐴)
9		simprr 771	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = (𝐹‘𝑤))
10		fveq2 6804	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
11		sbcie2s.b	. . . . . . . 8 ⊢ 𝐵 = (𝐹‘𝑊)
12	10, 11	eqtr4di 2794	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵)
13	12	adantr 482	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐹‘𝑤) = 𝐵)
14	9, 13	eqtrd 2776	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = 𝐵)
15		sbcie2s.1	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓))
16	8, 14, 15	syl2anc 585	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜑 ↔ 𝜓))
17	16	bicomd 222	. . 3 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜓 ↔ 𝜑))
18	17	ex 414	. 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜓 ↔ 𝜑)))
19	1, 2, 18	sbc2iedv 3806	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539  [wsbc 3721  ‘cfv 6458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-nul 5239

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6410  df-fv 6466

This theorem is referenced by:  istrkgc  26860  istrkgb  26861  istrkge  26863  istrkgl  26864  ishpg  27165  iscgra  27215