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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 4889 | . 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 1537 〈cotp 4639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 |
This theorem is referenced by: oteq123d 4893 idafval 18111 coafval 18118 arwlid 18126 arwrid 18127 arwass 18128 efgi 19752 efgtf 19755 efgtval 19756 efgval2 19757 mapdh6bN 41720 mapdh6cN 41721 mapdh6dN 41722 mapdh6gN 41725 hdmap1l6b 41794 hdmap1l6c 41795 hdmap1l6d 41796 hdmap1l6g 41799 |
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