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Theorem ifnefalse 3453
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 3450 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
ifnefalse (𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)

Proof of Theorem ifnefalse
StepHypRef Expression
1 df-ne 2284 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 iffalse 3450 . 2 𝐴 = 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 120 1 (𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1314  wne 2283  ifcif 3442
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 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ne 2284  df-if 3443
This theorem is referenced by:  xnegmnf  9563  rexneg  9564  xaddpnf1  9580  xaddpnf2  9581  xaddmnf1  9582  xaddmnf2  9583  mnfaddpnf  9585  rexadd  9586  fztpval  9814
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