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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 4351 | . 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 1475 〈cotp 4133 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 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-ot 4134 |
This theorem is referenced by: oteq123d 4355 idafval 16530 coafval 16537 arwlid 16545 arwrid 16546 arwass 16547 efgi 17955 efgtf 17958 efgtval 17959 efgval2 17960 mapdh6bN 36044 mapdh6cN 36045 mapdh6dN 36046 mapdh6gN 36049 hdmap1l6b 36119 hdmap1l6c 36120 hdmap1l6d 36121 hdmap1l6g 36124 |
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