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Mirrors > Home > ILE Home > Th. List > uneq2d | GIF version |
Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
uneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
uneq2d | ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | uneq2 3091 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∪ cun 2915 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 |
This theorem is referenced by: ifeq2 3335 tpeq3 3458 iununir 3738 unisucg 4151 relcoi1 4849 resasplitss 5069 fvun1 5239 fmptapd 5354 fvunsng 5357 rdgeq1 5958 rdgivallem 5968 rdgisuc1 5971 rdg0 5974 oav2 6043 oasuc 6044 omv2 6045 omsuc 6051 fzsuc 8931 fseq1p1m1 8956 fseq1m1p1 8957 fzosplitsnm1 9065 fzosplitsn 9089 fzosplitprm1 9090 |
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