Theorem ifnefalse 3531

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 3528 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
ifnefalse	|- ( A =/= B -> if ( A = B , C , D ) = D )

Proof of Theorem ifnefalse

Step	Hyp	Ref	Expression
1		df-ne 2337	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3528	. 2 |- ( -. A = B -> if ( A = B , C , D ) = D )
3	1, 2	sylbi 120	1 |- ( A =/= B -> if ( A = B , C , D ) = D )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   =/= wne 2336   ifcif 3520

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ne 2337  df-if 3521

This theorem is referenced by:  xnegmnf  9765  rexneg  9766  xaddpnf1  9782  xaddpnf2  9783  xaddmnf1  9784  xaddmnf2  9785  mnfaddpnf  9787  rexadd  9788  fztpval  10018  pcval  12228  lgsval3  13559  lgsdinn0  13589