Theorem ifnefalse 3637

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 3634 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 2415	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3634	. 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 2414   ifcif 3624

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 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ne 2415  df-if 3625

This theorem is referenced by:  xnegmnf  10181  rexneg  10182  xaddpnf1  10198  xaddpnf2  10199  xaddmnf1  10200  xaddmnf2  10201  mnfaddpnf  10203  rexadd  10204  fztpval  10439  pcval  13019  xpsfrnel  13608  znf1o  14925  znfi  14929  znhash  14930  lgsval3  16017  lgsdinn0  16047