sbcopeq1a - Metamath Proof Explorer

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Theorem sbcopeq1a 8055

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3758 that avoids the existential quantifiers of copsexg 5479). (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 3462	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3462	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 8006	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2772	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		sbceq1a 3758	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → (𝜑 ↔ [(2^nd ‘𝐴) / 𝑦]𝜑))
6	4, 5	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [(2^nd ‘𝐴) / 𝑦]𝜑))
7	1, 2	op1std 8005	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2772	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		sbceq1a 3758	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → ([(2^nd ‘𝐴) / 𝑦]𝜑 ↔ [(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑))
10	8, 9	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2^nd ‘𝐴) / 𝑦]𝜑 ↔ [(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑))
11	6, 10	bitr2d 283	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1^st ‘𝐴) / 𝑥][(2^nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1570 [wsbc 3747 ⟨cop 4600 ‘cfv 6543 1^st c1st 7993 2^nd c2nd 7994

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6499 df-fun 6545 df-fv 6551 df-1st 7995 df-2nd 7996

This theorem is used by: dfopab2 8058 dfoprab3s 8059 ralxpes 8141 frpoins3xpg 8145

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