Theorem ifnefalse 3568

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 3565 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 2365	. 2 |- ( A =/= B <-> -. A = B )
2		iffalse 3565	. 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 1364   =/= wne 2364   ifcif 3557

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ne 2365  df-if 3558

This theorem is referenced by:  xnegmnf  9895  rexneg  9896  xaddpnf1  9912  xaddpnf2  9913  xaddmnf1  9914  xaddmnf2  9915  mnfaddpnf  9917  rexadd  9918  fztpval  10149  pcval  12434  xpsfrnel  12927  znf1o  14139  znfi  14143  znhash  14144  lgsval3  15134  lgsdinn0  15164