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Theorem csbopeq1a 8046

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analogue of csbeq1a 3866). (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 3457	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7996	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2767	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		csbeq1a 3866	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
6	4, 5	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
7	1, 2	op1std 7995	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2767	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		csbeq1a 3866	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
10	8, 9	syl 18	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
11	6, 10	eqtr2d 2797	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 1568 ⦋csb 3852 ⟨cop 4594 ‘cfv 6536 1^st c1st 7983 2^nd c2nd 7984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 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 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-iota 6492 df-fun 6538 df-fv 6544 df-1st 7985 df-2nd 7986

This theorem is referenced by: dfmpo 8096 f1od2 33030 wdom2d2 43732

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