Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > tpeq3 | 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 3582 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq2d 3272 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐶, 𝐷} ∪ {𝐴}) = ({𝐶, 𝐷} ∪ {𝐵})) |
3 | df-tp 3579 | . 2 ⊢ {𝐶, 𝐷, 𝐴} = ({𝐶, 𝐷} ∪ {𝐴}) | |
4 | df-tp 3579 | . 2 ⊢ {𝐶, 𝐷, 𝐵} = ({𝐶, 𝐷} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∪ cun 3110 {csn 3571 {cpr 3572 {ctp 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-sn 3577 df-tp 3579 |
This theorem is referenced by: tpeq3d 3662 tppreq3 3674 fztpval 10009 |
Copyright terms: Public domain | W3C validator |