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Mirrors > Home > NFE 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 3412 | . 2 ⊢ (A = B → (C ∪ A) = (C ∪ B)) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → (C ∪ A) = (C ∪ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∪ cun 3207 |
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-nan 1288 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-v 2861 df-nin 3211 df-compl 3212 df-un 3214 |
This theorem is referenced by: ifeq2 3667 tpeq3 3810 iununi 4050 prprc2 4122 pw1eqadj 4332 elsuci 4414 nnsucelr 4428 nnadjoin 4520 opeq2 4579 fvun1 5379 fvunsn 5444 enadj 6060 nmembers1lem3 6270 frecxp 6314 |
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