Theorem sbcie2s 16843

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 6781	. 2 ⊢ (𝐸‘𝑤) ∈ V
2		fvex 6781	. 2 ⊢ (𝐹‘𝑤) ∈ V
3		simprl 767	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = (𝐸‘𝑤))
4		fveq2 6768	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5		sbcie2s.a	. . . . . . . 8 ⊢ 𝐴 = (𝐸‘𝑊)
6	4, 5	eqtr4di 2797	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐸‘𝑤) = 𝐴)
8	3, 7	eqtrd 2779	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = 𝐴)
9		simprr 769	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = (𝐹‘𝑤))
10		fveq2 6768	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
11		sbcie2s.b	. . . . . . . 8 ⊢ 𝐵 = (𝐹‘𝑊)
12	10, 11	eqtr4di 2797	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵)
13	12	adantr 480	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐹‘𝑤) = 𝐵)
14	9, 13	eqtrd 2779	. . . . 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 583	. . . 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 412	. 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜓 ↔ 𝜑)))
19	1, 2, 18	sbc2iedv 3805	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541  [wsbc 3719  ‘cfv 6430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438

This theorem is referenced by:  istrkgc  26796  istrkgb  26797  istrkge  26799  istrkgl  26800  ishpg  27101  iscgra  27151