Theorem csbopeq1a 8035

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analogue of csbeq1a 3902). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

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

Proof of Theorem csbopeq1a

Step	Hyp	Ref	Expression
1		vex 3472	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3472	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7985	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = 𝑦)
4	3	eqcomd 2732	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝐴))
5		csbeq1a 3902	. . 3 ⊢ (𝑦 = (2nd ‘𝐴) → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
7	1, 2	op1std 7984	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = 𝑥)
8	7	eqcomd 2732	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st ‘𝐴))
9		csbeq1a 3902	. . 3 ⊢ (𝑥 = (1st ‘𝐴) → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
10	8, 9	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
11	6, 10	eqtr2d 2767	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1533  ⦋csb 3888  ⟨cop 4629  ‘cfv 6537  1^st c1st 7972  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-1st 7974  df-2nd 7975

This theorem is referenced by:  dfmpo  8088  f1od2  32453  wdom2d2  42352