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Mirrors > Home > MPE Home > Th. List > tpeq2d | Structured version Visualization version GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Ref | Expression |
---|---|
tpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
tpeq2d | ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | tpeq2 4673 | . 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {ctp 4565 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3941 df-sn 4562 df-pr 4564 df-tp 4566 |
This theorem is referenced by: tpeq123d 4678 fntpb 6966 erngset 37930 erngset-rN 37938 |
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