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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 (analogue of sbceq1a 3798 that avoids the existential quantifiers of copsexg 5495). (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 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | vex 3483 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | op2ndd 8026 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = 𝑦) | 
| 4 | 3 | eqcomd 2742 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝐴)) | 
| 5 | sbceq1a 3798 | . . 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 8025 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = 𝑥) | 
| 8 | 7 | eqcomd 2742 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → 𝑥 = (1st ‘𝐴)) | 
| 9 | sbceq1a 3798 | . . 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 280 | 1 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 [wsbc 3787 〈cop 4631 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fv 6568 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: dfopab2 8078 dfoprab3s 8079 ralxpes 8162 frpoins3xpg 8166 | 
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