sbcopeq1a - Metamath Proof Explorer

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Theorem sbcopeq1a 8060

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3749 that avoids the existential quantifiers of copsexg 5461). (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 3454	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 8012	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2766	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		sbceq1a 3749	. . 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 8011	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2766	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		sbceq1a 3749	. . 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 1569 [wsbc 3738 ⟨cop 4834 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6540 df-fv 6546 df-1st 8001 df-2nd 8002

This theorem is used by: dfopab2 8063 dfoprab3s 8064 ralxpes 8155 frpoins3xpg 8159

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