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Mirrors > Home > ILE Home > Th. List > uneq1d | GIF version |
Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
uneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
uneq1d | ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | uneq1 3264 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∪ cun 3109 |
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 2723 df-un 3115 |
This theorem is referenced by: ifeq1 3518 preq1 3647 tpeq1 3656 tpeq2 3657 resasplitss 5361 fmptpr 5671 funresdfunsnss 5682 rdgisucinc 6344 oasuc 6423 omsuc 6431 funresdfunsndc 6465 fisseneq 6888 sbthlemi5 6917 exmidfodomrlemim 7148 fzpred 9995 fseq1p1m1 10019 nn0split 10061 nnsplit 10062 fzo0sn0fzo1 10146 fzosplitprm1 10159 fsum1p 11345 fprod1p 11526 zsupcllemstep 11863 setsvala 12368 setsabsd 12376 setscom 12377 |
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