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Mirrors > Home > ILE Home > Th. List > unieqd | GIF version |
Description: Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.) |
Ref | Expression |
---|---|
unieqd.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
unieqd | ⊢ (φ → ∪ A = ∪ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieqd.1 | . 2 ⊢ (φ → A = B) | |
2 | unieq 3580 | . 2 ⊢ (A = B → ∪ A = ∪ B) | |
3 | 1, 2 | syl 14 | 1 ⊢ (φ → ∪ A = ∪ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∪ cuni 3571 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-uni 3572 |
This theorem is referenced by: uniprg 3586 unisng 3588 unisn3 4146 opswapg 4750 elxp4 4751 elxp5 4752 iotaeq 4818 iotabi 4819 uniabio 4820 funfvdm 5179 funfvdm2 5180 fvun1 5182 fniunfv 5344 funiunfvdm 5345 1stvalg 5711 2ndvalg 5712 fo1st 5726 fo2nd 5727 f1stres 5728 f2ndres 5729 2nd1st 5748 cnvf1olem 5787 brtpos2 5807 dftpos4 5819 tpostpos 5820 recseq 5862 tfrexlem 5889 xpcomco 6236 xpassen 6240 xpdom2 6241 |
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