Theorem sbcopeq1a 6078

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2913 that avoids the existential quantifiers of copsexg 4161). (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 2684	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 2684	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 6040	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝐴) = 𝑦)
4	3	eqcomd 2143	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2nd ‘𝐴))
5		sbceq1a 2913	. . 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 6039	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = 𝑥)
8	7	eqcomd 2143	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1st ‘𝐴))
9		sbceq1a 2913	. . 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 1331  [wsbc 2904  ⟨cop 3525  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by:  dfopab2  6080  dfoprab3s  6081