Theorem sbcopeq1a 8053

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3781 that avoids the existential quantifiers of copsexg 5471). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
sbcopeq1a	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Proof of Theorem sbcopeq1a

Step	Hyp	Ref	Expression
1		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3468	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 8004	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = 𝑦)
4	3	eqcomd 2742	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝐴))
5		sbceq1a 3781	. . 3 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑))
7	1, 2	op1std 8003	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = 𝑥)
8	7	eqcomd 2742	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st ‘𝐴))
9		sbceq1a 3781	. . 3 ⊢ (𝑥 = (1st ‘𝐴) → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑))
10	8, 9	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑))
11	6, 10	bitr2d 280	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  [wsbc 3770  ⟨cop 4612  ‘cfv 6536  1^st c1st 7991  2^nd c2nd 7992

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-1st 7993  df-2nd 7994

This theorem is referenced by:  dfopab2  8056  dfoprab3s  8057  ralxpes  8140  frpoins3xpg  8144