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Mirrors > Home > MPE Home > Th. List > tpeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
tpeq3 | ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4641 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq2d 4178 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵})) |
3 | df-tp 4636 | . 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴}) | |
4 | df-tp 4636 | . 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∪ cun 3961 {csn 4631 {cpr 4633 {ctp 4635 |
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-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-tp 4636 |
This theorem is referenced by: tpeq3d 4752 tppreq3 4764 fntpb 7229 fztpval 13623 hashtpg 14521 dvh4dimN 41430 grimgrtri 47852 usgrgrtrirex 47853 grlimgrtri 47899 usgrexmpl1tri 47920 |
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