Theorem sbcie3s 16863

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 6789	. 2 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) ∈ V)
2		fvexd 6789	. . 3 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → (𝐹‘𝑤) ∈ V)
3		fvexd 6789	. . . 4 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → (𝐺‘𝑤) ∈ V)
4		simpllr 773	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = (𝐸‘𝑤))
5		fveq2 6774	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
6	5	ad3antrrr 727	. . . . . . . 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 2796	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑎 = 𝐴)
10		simplr 766	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = (𝐹‘𝑤))
11		fveq2 6774	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
12	11	ad3antrrr 727	. . . . . . . 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 2796	. . . . . 6 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑏 = 𝐵)
16		simpr 485	. . . . . . . 8 ⊢ ((((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) ∧ 𝑐 = (𝐺‘𝑤)) → 𝑐 = (𝐺‘𝑤))
17		fveq2 6774	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝐺‘𝑤) = (𝐺‘𝑊))
18	17	ad3antrrr 727	. . . . . . . 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 2796	. . . . . 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 1370	. . . . 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 3761	. . 3 ⊢ (((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) ∧ 𝑏 = (𝐹‘𝑤)) → ([(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
26	2, 25	sbcied 3761	. 2 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = (𝐸‘𝑤)) → ([(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))
27	1, 26	sbcied 3761	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏][(𝐺‘𝑤) / 𝑐]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  Vcvv 3432  [wsbc 3716  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  istrkgcb  26817  istrkgld  26820  legval  26945  istrkg2d  32646  afsval  32651