Theorem neeq1d 2421

Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Hypothesis

Ref	Expression
neeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
neeq1d	|- ( ph -> ( A =/= C <-> B =/= C ) )

Proof of Theorem neeq1d

Step	Hyp	Ref	Expression
1		neeq1d.1	. 2 |- ( ph -> A = B )
2		neeq1 2416	. 2 |- ( A = B -> ( A =/= C <-> B =/= C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A =/= C <-> B =/= C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   =/= wne 2403

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-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2404

This theorem is referenced by:  neeq12d  2423  eqnetrd  2427  prnzg  3801  suppval1  6417  elsuppfng  6420  elsuppfn  6421  suppsnopdc  6428  ressuppss  6432  pw2f1odclem  7063  hashprg  11118  algcvg  12683  algcvga  12686  eucalgcvga  12693  rpdvds  12734  phibndlem  12851  dfphi2  12855  pcaddlem  12975  ennnfoneleminc  13095  ennnfonelemex  13098  ennnfonelemhom  13099  ennnfonelemnn0  13106  ennnfonelemr  13107  ennnfonelemim  13108  ctinfomlemom  13111  setscomd  13186  rrgsupp  14344  pellexlem3  15776  lgsne0  15840  umgr2cwwkdifex  16349  dceqnconst  16776  dcapnconst  16777  nconstwlpolem  16781