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Mirrors > Home > NFE Home > Th. List > ifnefalse | GIF version |
Description: When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3669 directly in this case. It happens, e.g., in oevn0 in set.mm. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
ifnefalse | ⊢ (A ≠ B → if(A = B, C, D) = D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2518 | . 2 ⊢ (A ≠ B ↔ ¬ A = B) | |
2 | iffalse 3669 | . 2 ⊢ (¬ A = B → if(A = B, C, D) = D) | |
3 | 1, 2 | sylbi 187 | 1 ⊢ (A ≠ B → if(A = B, C, D) = D) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ≠ wne 2516 ifcif 3662 |
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-ne 2518 df-if 3663 |
This theorem is referenced by: (None) |
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