Theorem ifnefalse 3633

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 3630 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 2413	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3630	. 2 |- ( -. A = B -> if ( A = B , C , D ) = D )
3	1, 2	sylbi 121	1 |- ( A =/= B -> if ( A = B , C , D ) = D )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   =/= wne 2412   ifcif 3620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ne 2413  df-if 3621

This theorem is referenced by:  xnegmnf  10162  rexneg  10163  xaddpnf1  10179  xaddpnf2  10180  xaddmnf1  10181  xaddmnf2  10182  mnfaddpnf  10184  rexadd  10185  fztpval  10417  pcval  12994  xpsfrnel  13557  znf1o  14799  znfi  14803  znhash  14804  lgsval3  15891  lgsdinn0  15921