Theorem sbcie3s 16791

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 6771	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		fvexd 6771	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝐹‘𝑤) ∈ V)
3		fvexd 6771	. . . 4 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → (𝐺‘𝑤) ∈ V)
4		simpllr 772	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = (𝐸‘𝑤))
5		fveq2 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
6	5	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐸‘𝑤) = (𝐸‘𝑊))
7	4, 6	eqtrd 2778	. . . . . . 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 2797	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = 𝐴)
10		simplr 765	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = (𝐹‘𝑤))
11		fveq2 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
12	11	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐹‘𝑤) = (𝐹‘𝑊))
13	10, 12	eqtrd 2778	. . . . . . 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 2797	. . . . . 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 6756	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐺‘𝑤) = (𝐺‘𝑊))
18	17	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝐺‘𝑤) = (𝐺‘𝑊))
19	16, 18	eqtrd 2778	. . . . . . 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 2797	. . . . . 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 1369	. . . . 5 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝜑 ↔ 𝜓))
24	23	bicomd 222	. . . 4 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → (𝜓 ↔ 𝜑))
25	3, 24	sbcied 3756	. . 3 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → ([(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
26	2, 25	sbcied 3756	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → ([(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
27	1, 26	sbcied 3756	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  Vcvv 3422  [wsbc 3711  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426

This theorem is referenced by:  istrkgcb  26721  istrkgld  26724  legval  26849  istrkg2d  32546  afsval  32551