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Theorem sbcopeq1a 7751

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3786 that avoids the existential quantifiers of copsexg 5385). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

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

**Proof of Theorem sbcopeq1a**
Step	Hyp	Ref	Expression
1		vex 3500	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3500	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7703	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2830	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		sbceq1a 3786	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → (𝜑 ↔ [(2^nd ‘𝐴) / 𝑦]𝜑))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [(2^nd ‘𝐴) / 𝑦]𝜑))
7	1, 2	op1std 7702	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2830	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		sbceq1a 3786	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → ([(2^nd ‘𝐴) / 𝑦]𝜑 ↔ [(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑))
10	8, 9	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2^nd ‘𝐴) / 𝑦]𝜑 ↔ [(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑))
11	6, 10	bitr2d 282	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 [wsbc 3775 ⟨cop 4576 ‘cfv 6358 1^st c1st 7690 2^nd c2nd 7691

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fv 6366 df-1st 7692 df-2nd 7693

This theorem is referenced by: dfopab2 7753 dfoprab3s 7754

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