Theorem csbopeq1a 5814

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 2860). (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 2560	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 2560	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 5776	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = 𝑦)
4	3	eqcomd 2045	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝐴))
5		csbeq1a 2860	. . 3 ⊢ (𝑦 = (2nd ‘𝐴) → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
6	4, 5	syl 14	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = ⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
7	1, 2	op1std 5775	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = 𝑥)
8	7	eqcomd 2045	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st ‘𝐴))
9		csbeq1a 2860	. . 3 ⊢ (𝑥 = (1st ‘𝐴) → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
10	8, 9	syl 14	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵)
11	6, 10	eqtr2d 2073	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1243  ⦋csb 2852  ⟨cop 3378  ‘cfv 4902  1^st c1st 5765  2^nd c2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by:  dfmpt2  5844