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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 3223 | . 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 1331 ∪ cun 3069 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 |
This theorem is referenced by: ifeq1 3477 preq1 3600 tpeq1 3609 tpeq2 3610 resasplitss 5302 fmptpr 5612 funresdfunsnss 5623 rdgisucinc 6282 oasuc 6360 omsuc 6368 funresdfunsndc 6402 fisseneq 6820 sbthlemi5 6849 exmidfodomrlemim 7057 fzpred 9850 fseq1p1m1 9874 nn0split 9913 nnsplit 9914 fzo0sn0fzo1 9998 fzosplitprm1 10011 fsum1p 11187 zsupcllemstep 11638 setsvala 11990 setsabsd 11998 setscom 11999 |
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