Theorem ifnefalse 3651

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 3648 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 2421	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3648	. 2 |- ( -. A = B -> if ( A = B , C , D ) = D )
3	1, 2	sylbi 121	1 |- ( A =/= B -> if ( A = B , C , D ) = D )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ne 2421  df-if 3639

This theorem is used by:  xnegmnf  10247  rexneg  10248  xaddpnf1  10264  xaddpnf2  10265  xaddmnf1  10266  xaddmnf2  10267  mnfaddpnf  10269  rexadd  10270  fztpval  10506  pcval  13103  xpsfrnel  13723  znf1o  15094  znfi  15098  znhash  15099  lgsval3  16359  lgsdinn0  16389