Theorem neeq1d 2438

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 2433	. 2 |- ( A = B -> ( A =/= C <-> B =/= C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A =/= C <-> B =/= C ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  neeq12d  2440  eqnetrd  2444  prnzg  3838  suppval1  6483  elsuppfng  6486  elsuppfn  6487  suppsnopdc  6494  ressuppss  6498  pw2f1odclem  7138  hashprg  11271  algcvg  12851  algcvga  12854  eucalgcvga  12861  rpdvds  12902  phibndlem  13023  dfphi2  13027  pcaddlem  13147  ennnfoneleminc  13360  ennnfonelemex  13363  ennnfonelemhom  13364  ennnfonelemnn0  13371  ennnfonelemr  13372  ennnfonelemim  13373  ctinfomlemom  13376  setscomd  13451  rrgsupp  14683  pellexlem3  16231  lgsne0  16362  umgr2cwwkdifex  16871  dceqnconst  17317  dcapnconst  17318  nconstwlpolem  17322