Theorem necomd 2488

Description: Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)

Hypothesis

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

Assertion

Ref	Expression
necomd	|- ( ph -> B =/= A )

Proof of Theorem necomd

Step	Hyp	Ref	Expression
1		necomd.1	. 2 |- ( ph -> A =/= B )
2		necom 2486	. 2 |- ( A =/= B <-> B =/= A )
3	1, 2	sylib 122	1 |- ( ph -> B =/= A )

Syntax hints:   -> wi 4   =/= wne 2402

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 1495  ax-gen 1497  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-ne 2403

This theorem is referenced by:  ifnefals  3650  difsnb  3816  0nelop  4340  frecabcl  6565  fidifsnen  7057  tpfidisj  7121  omp1eomlem  7293  difinfsnlem  7298  fodjuomnilemdc  7343  en2eleq  7406  en2other2  7407  netap  7473  2omotaplemap  7476  ltned  8293  lt0ne0  8608  zdceq  9555  zneo  9581  xrlttri3  10032  qdceq  10505  flqltnz  10548  seqf1oglem1  10782  nn0opthd  10985  hashdifpr  11085  hashtpgim  11110  cats1un  11306  sumtp  11993  nninfctlemfo  12629  isprm2lem  12706  oddprm  12850  pcmpt  12934  ennnfonelemex  13053  perfectlem2  15743  lgsneg  15772  lgseisenlem4  15821  lgsquadlem1  15825  lgsquadlem3  15827  lgsquad2  15831  2lgsoddprm  15861  funvtxval0d  15903  umgrvad2edg  16081  1hegrvtxdg1rfi  16180  vdegp1bid  16185  umgr2cwwk2dif  16294  eupth2lem3lem4fi  16343  pw1ndom3lem  16639  pw1ndom3  16640