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Mirrors > Home > MPE Home > Th. List > sbcopeq1a | Structured version Visualization version GIF version |
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3413 that avoids the existential quantifiers of copsexg 4882). (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 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | op2ndd 7070 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) |
4 | 3 | eqcomd 2616 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) |
5 | sbceq1a 3413 | . . 3 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ [(2nd ‘𝐴) / 𝑦]𝜑)) |
7 | 1, 2 | op1std 7069 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) |
8 | 7 | eqcomd 2616 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) |
9 | sbceq1a 3413 | . . 3 ⊢ (𝑥 = (1st ‘𝐴) → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(2nd ‘𝐴) / 𝑦]𝜑 ↔ [(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑)) |
11 | 6, 10 | bitr2d 268 | 1 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 [wsbc 3402 〈cop 4131 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: dfopab2 7113 dfoprab3s 7114 |
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