Theorem ifnefalse 3616

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 3613 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 2403	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3613	. 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 1397   =/= wne 2402   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ne 2403  df-if 3606

This theorem is referenced by:  xnegmnf  10063  rexneg  10064  xaddpnf1  10080  xaddpnf2  10081  xaddmnf1  10082  xaddmnf2  10083  mnfaddpnf  10085  rexadd  10086  fztpval  10317  pcval  12868  xpsfrnel  13426  znf1o  14664  znfi  14668  znhash  14669  lgsval3  15746  lgsdinn0  15776