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Theorem csbopeq1a 8016

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analogue of csbeq1a 3857). (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 3448	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3448	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7966	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2758	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		csbeq1a 3857	. . 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 7965	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2758	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		csbeq1a 3857	. . 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 2788	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 1550 ⦋csb 3843 ⟨cop 4578 ‘cfv 6506 1^st c1st 7953 2^nd c2nd 7954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-iota 6462 df-fun 6508 df-fv 6514 df-1st 7955 df-2nd 7956

This theorem is referenced by: dfmpo 8065 f1od2 32860 wdom2d2 43550

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