csbopeq1a - Metamath Proof Explorer

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Theorem csbopeq1a 8020

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 3452	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3452	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7970	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2762	. . 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 7969	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2762	. . 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 2792	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 1554 ⦋csb 3847 ⟨cop 4582 ‘cfv 6510 1^st c1st 7957 2^nd c2nd 7958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pr 5384 ax-un 7707

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-iota 6466 df-fun 6512 df-fv 6518 df-1st 7959 df-2nd 7960

This theorem is referenced by: dfmpo 8069 f1od2 32864 wdom2d2 43560

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