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Mirrors > Home > ILE Home > Th. List > preq1 | GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
preq1 | ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3543 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
2 | 1 | uneq1d 3234 | . 2 ⊢ (𝐴 = 𝐵 → ({𝐴} ∪ {𝐶}) = ({𝐵} ∪ {𝐶})) |
3 | df-pr 3539 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
4 | df-pr 3539 | . 2 ⊢ {𝐵, 𝐶} = ({𝐵} ∪ {𝐶}) | |
5 | 2, 3, 4 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∪ cun 3074 {csn 3532 {cpr 3533 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: preq2 3609 preq12 3610 preq1i 3611 preq1d 3614 tpeq1 3617 prnzg 3655 preq12b 3705 preq12bg 3708 opeq1 3713 uniprg 3759 intprg 3812 prexg 4141 opthreg 4479 bdxmet 12709 bj-prexg 13280 |
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