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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 3795 | . 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 1342 ∪ cuni 3786 |
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-rex 2448 df-uni 3787 |
This theorem is referenced by: uniprg 3801 unisng 3803 unisn3 4420 onsucuni2 4538 opswapg 5087 elxp4 5088 elxp5 5089 iotaeq 5158 iotabi 5159 uniabio 5160 funfvdm 5546 funfvdm2 5547 fvun1 5549 fniunfv 5727 funiunfvdm 5728 1stvalg 6105 2ndvalg 6106 fo1st 6120 fo2nd 6121 f1stres 6122 f2ndres 6123 2nd1st 6143 cnvf1olem 6186 brtpos2 6213 dftpos4 6225 tpostpos 6226 recseq 6268 tfrexlem 6296 ixpsnf1o 6696 xpcomco 6786 xpassen 6790 xpdom2 6791 supeq1 6945 supeq2 6948 supeq3 6949 supeq123d 6950 en2other2 7146 dfinfre 8845 hashinfom 10685 hashennn 10687 fsumcnv 11372 fprodcnv 11560 isbasisg 12640 basis1 12643 baspartn 12646 tgval 12647 eltg 12650 ntrfval 12698 ntrval 12708 tgrest 12767 restuni2 12775 lmfval 12790 cnfval 12792 cnpfval 12793 txtopon 12860 txswaphmeolem 12918 peano4nninf 13779 |
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