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Mirrors > Home > NFE 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 3900 | . 2 ⊢ (A = B → ∪A = ∪B) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → ∪A = ∪B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∪cuni 3891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-uni 3892 |
This theorem is referenced by: uniprg 3906 unisng 3908 iotaeq 4347 iotabi 4348 uniabio 4349 iotanul 4354 dfiota4 4372 elxp4 5108 funfv 5375 fvun 5378 fvco2 5382 fniunfv 5466 |
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