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| Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) | 
| Ref | Expression | 
|---|---|
| tpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| tpeq3d | ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | tpeq3 4744 | . 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 1540 {ctp 4630 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-tp 4631 | 
| This theorem is referenced by: tpeq123d 4748 fntpb 7229 erngset 40802 erngset-rN 40810 | 
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