Theorem necomd 2500

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 2498	. 2 |- ( A =/= B <-> B =/= A )
3	1, 2	sylib 122	1 |- ( ph -> B =/= A )

Syntax hints:   -> wi 4   =/= wne 2414

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-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-ne 2415

This theorem is referenced by:  ifnefals  3671  difsnb  3842  0nelop  4369  frecabcl  6643  fidifsnen  7138  tpfidisj  7202  omp1eomlem  7398  difinfsnlem  7403  fodjuomnilemdc  7448  en2eleq  7511  en2other2  7512  netap  7584  2omotaplemap  7587  ltned  8403  lt0ne0  8719  zdceq  9670  zneo  9697  xrlttri3  10149  qdceq  10628  flqltnz  10671  seqf1oglem1  10905  nn0opthd  11109  hashdifpr  11210  hashtpgim  11242  cats1un  11438  sumtp  12125  nninfctlemfo  12761  isprm2lem  12838  oddprm  12982  pcmpt  13066  ennnfonelemex  13249  perfectlem2  15994  lgsneg  16023  lgseisenlem4  16072  lgsquadlem1  16076  lgsquadlem3  16078  lgsquad2  16082  2lgsoddprm  16112  funvtxval0d  16154  umgrvad2edg  16332  1hegrvtxdg1rfi  16431  vdegp1bid  16436  umgr2cwwk2dif  16545  eupth2lem3lem4fi  16594  pw1ndom3lem  16889  pw1ndom3  16890