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Mirrors > Home > ILE Home > Th. List > sbcopeq1a | GIF version |
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2964 that avoids the existential quantifiers of copsexg 4227). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
sbcopeq1a | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 2733 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | op2ndd 6125 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) |
4 | 3 | eqcomd 2176 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) |
5 | sbceq1a 2964 | . . 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 6124 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) |
8 | 7 | eqcomd 2176 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) |
9 | sbceq1a 2964 | . . 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)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 [wsbc 2955 〈cop 3584 ‘cfv 5196 1st c1st 6114 2nd c2nd 6115 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fv 5204 df-1st 6116 df-2nd 6117 |
This theorem is referenced by: dfopab2 6165 dfoprab3s 6166 |
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