Theorem ifnefalse 3614

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 3611 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 2401	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3611	. 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 1395   =/= wne 2400   ifcif 3603

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 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ne 2401  df-if 3604

This theorem is referenced by:  xnegmnf  10054  rexneg  10055  xaddpnf1  10071  xaddpnf2  10072  xaddmnf1  10073  xaddmnf2  10074  mnfaddpnf  10076  rexadd  10077  fztpval  10308  pcval  12859  xpsfrnel  13417  znf1o  14655  znfi  14659  znhash  14660  lgsval3  15737  lgsdinn0  15767