sbcopeq1a - Metamath Proof Explorer

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Theorem sbcopeq1a 8049

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3750 that avoids the existential quantifiers of copsexg 5462). (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 3455	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3455	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 8001	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2767	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		sbceq1a 3750	. . 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 8000	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2767	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		sbceq1a 3750	. . 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 3739 ⟨cop 4590 ‘cfv 6531 1^st c1st 7988 2^nd c2nd 7989

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7740

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6487 df-fun 6533 df-fv 6539 df-1st 7990 df-2nd 7991

This theorem is used by: dfopab2 8052 dfoprab3s 8053 ralxpes 8137 frpoins3xpg 8141

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