Theorem ifnefalse 3451

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 3448 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 2283	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3448	. 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 1314   =/= wne 2282   ifcif 3440

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 2283  df-if 3441

This theorem is referenced by:  xnegmnf  9505  rexneg  9506  xaddpnf1  9522  xaddpnf2  9523  xaddmnf1  9524  xaddmnf2  9525  mnfaddpnf  9527  rexadd  9528  fztpval  9756