Theorem sbcie3s 17181

Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
sbcie3s.a	⊢ 𝐴 = (𝐸‘𝑊)
sbcie3s.b	⊢ 𝐵 = (𝐹‘𝑊)
sbcie3s.c	⊢ 𝐶 = (𝐺‘𝑊)
sbcie3s.1	⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcie3s	⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))

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

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

Proof of Theorem sbcie3s

Step	Hyp	Ref	Expression
1		fvexd 6891	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		fvexd 6891	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝐹‘𝑤) ∈ V)
3		fvexd 6891	. . . 4 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → (𝐺‘𝑤) ∈ V)
4		simpllr 775	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = (𝐸‘𝑤))
5		fveq2 6876	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
6	5	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐸‘𝑤) = (𝐸‘𝑊))
7	4, 6	eqtrd 2770	. . . . . . 7 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = (𝐸‘𝑊))
8		sbcie3s.a	. . . . . . 7 ⊢ 𝐴 = (𝐸‘𝑊)
9	7, 8	eqtr4di 2788	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = 𝐴)
10		simplr 768	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = (𝐹‘𝑤))
11		fveq2 6876	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
12	11	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐹‘𝑤) = (𝐹‘𝑊))
13	10, 12	eqtrd 2770	. . . . . . 7 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = (𝐹‘𝑊))
14		sbcie3s.b	. . . . . . 7 ⊢ 𝐵 = (𝐹‘𝑊)
15	13, 14	eqtr4di 2788	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = 𝐵)
16		simpr 484	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑐 = (𝐺‘𝑤))
17		fveq2 6876	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐺‘𝑤) = (𝐺‘𝑊))
18	17	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐺‘𝑤) = (𝐺‘𝑊))
19	16, 18	eqtrd 2770	. . . . . . 7 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑐 = (𝐺‘𝑊))
20		sbcie3s.c	. . . . . . 7 ⊢ 𝐶 = (𝐺‘𝑊)
21	19, 20	eqtr4di 2788	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑐 = 𝐶)
22		sbcie3s.1	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶) → (𝜑 ↔ 𝜓))
23	9, 15, 21, 22	syl3anc 1373	. . . . 5 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝜑 ↔ 𝜓))
24	23	bicomd 223	. . . 4 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝜓 ↔ 𝜑))
25	3, 24	sbcied 3809	. . 3 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → ([(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
26	2, 25	sbcied 3809	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → ([(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
27	1, 26	sbcied 3809	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Vcvv 3459  [wsbc 3765  ‘cfv 6531

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539

This theorem is referenced by:  istrkgcb  28435  istrkgld  28438  legval  28563  istrkg2d  34698  afsval  34703