Theorem sbcopeq1a 6155

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2960 that avoids the existential quantifiers of copsexg 4222). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
sbcopeq1a	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Proof of Theorem sbcopeq1a

Step	Hyp	Ref	Expression
1		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 2729	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 6117	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = 𝑦)
4	3	eqcomd 2171	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝐴))
5		sbceq1a 2960	. . 3 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑))
6	4, 5	syl 14	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑))
7	1, 2	op1std 6116	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = 𝑥)
8	7	eqcomd 2171	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st ‘𝐴))
9		sbceq1a 2960	. . 3 ⊢ (𝑥 = (1st ‘𝐴) → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑))
10	8, 9	syl 14	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑))
11	6, 10	bitr2d 188	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  [wsbc 2951  ⟨cop 3579  ‘cfv 5188  1^st c1st 6106  2^nd c2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  dfopab2  6157  dfoprab3s  6158