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| Mirrors > Home > MPE Home > Th. List > oteq3d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| oteq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| oteq3d | ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oteq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | oteq3 4883 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 〈cotp 4633 | 
| 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-ext 2707 | 
| 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-ot 4634 | 
| This theorem is referenced by: oteq123d 4887 idafval 18103 coafval 18110 arwlid 18118 arwrid 18119 arwass 18120 efgi 19738 efgtf 19741 efgtval 19742 efgval2 19743 mapdh6bN 41740 mapdh6cN 41741 mapdh6dN 41742 mapdh6gN 41745 hdmap1l6b 41814 hdmap1l6c 41815 hdmap1l6d 41816 hdmap1l6g 41819 | 
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