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Mirrors > Home > NFE Home > Th. List > nineq12i | GIF version |
Description: Equality inference for anti-intersection. (Contributed by SF, 11-Jan-2015.) |
Ref | Expression |
---|---|
nineqi.1 | ⊢ A = B |
nineq12i.2 | ⊢ C = D |
Ref | Expression |
---|---|
nineq12i | ⊢ (A ⩃ C) = (B ⩃ D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nineqi.1 | . 2 ⊢ A = B | |
2 | nineq12i.2 | . 2 ⊢ C = D | |
3 | nineq12 3236 | . 2 ⊢ ((A = B ∧ C = D) → (A ⩃ C) = (B ⩃ D)) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ (A ⩃ C) = (B ⩃ D) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ⩃ cnin 3204 |
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 |
This theorem is referenced by: dfin5 3545 |
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