Theorem sbcie2s 15916

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 6201	. 2 ⊢ (𝐸‘𝑤) ∈ V
2		fvex 6201	. 2 ⊢ (𝐹‘𝑤) ∈ V
3		simprl 794	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = (𝐸‘𝑤))
4		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = (𝐸‘𝑊))
5		sbcie2s.a	. . . . . . . 8 ⊢ 𝐴 = (𝐸‘𝑊)
6	4, 5	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐸‘𝑤) = 𝐴)
7	6	adantr 481	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐸‘𝑤) = 𝐴)
8	3, 7	eqtrd 2656	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑎 = 𝐴)
9		simprr 796	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → 𝑏 = (𝐹‘𝑤))
10		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = (𝐹‘𝑊))
11		sbcie2s.b	. . . . . . . 8 ⊢ 𝐵 = (𝐹‘𝑊)
12	10, 11	syl6eqr 2674	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐹‘𝑤) = 𝐵)
13	12	adantr 481	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝐹‘𝑤) = 𝐵)
14	9, 13	eqtrd 2656	. . . . 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 693	. . . 4 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜑 ↔ 𝜓))
17	16	bicomd 213	. . 3 ⊢ ((𝑤 = 𝑊 ∧ (𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤))) → (𝜓 ↔ 𝜑))
18	17	ex 450	. 2 ⊢ (𝑤 = 𝑊 → ((𝑎 = (𝐸‘𝑤) ∧ 𝑏 = (𝐹‘𝑤)) → (𝜓 ↔ 𝜑)))
19	1, 2, 18	sbc2iedv 3506	1 ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  [wsbc 3435  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  istrkgc  25353  istrkgb  25354  istrkge  25356  istrkgl  25357  ishpg  25651  iscgra  25701