![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE 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 versus applying iffalse 3487 directly in this case. (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 2310 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | iffalse 3487 | . 2 ⊢ (¬ 𝐴 = 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1332 ≠ wne 2309 ifcif 3479 |
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-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-ne 2310 df-if 3480 |
This theorem is referenced by: xnegmnf 9642 rexneg 9643 xaddpnf1 9659 xaddpnf2 9660 xaddmnf1 9661 xaddmnf2 9662 mnfaddpnf 9664 rexadd 9665 fztpval 9894 |
Copyright terms: Public domain | W3C validator |