csbopeq1a - Metamath Proof Explorer

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Theorem csbopeq1a 7991

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analogue of csbeq1a 3861). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Assertion**
Ref	Expression
csbopeq1a	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

**Proof of Theorem csbopeq1a**
Step	Hyp	Ref	Expression
1		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7941	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2739	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		csbeq1a 3861	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
7	1, 2	op1std 7940	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2739	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		csbeq1a 3861	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
10	8, 9	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
11	6, 10	eqtr2d 2769	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ⦋csb 3847 ⟨cop 4583 ‘cfv 6489 1^st c1st 7928 2^nd c2nd 7929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-iota 6445 df-fun 6491 df-fv 6497 df-1st 7930 df-2nd 7931

This theorem is referenced by: dfmpo 8041 f1od2 32713 wdom2d2 43142

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