Theorem pm5.21nii 711

Description: Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)

Hypotheses

Ref	Expression
pm5.21ni.1	|- ( ph -> ps )
pm5.21ni.2	|- ( ch -> ps )
pm5.21nii.3	|- ( ps -> ( ph <-> ch ) )

Assertion

Ref	Expression
pm5.21nii	|- ( ph <-> ch )

Proof of Theorem pm5.21nii

Step	Hyp	Ref	Expression
1		pm5.21ni.1	. . . 4 |- ( ph -> ps )
2		pm5.21nii.3	. . . 4 |- ( ps -> ( ph <-> ch ) )
3	1, 2	syl 14	. . 3 |- ( ph -> ( ph <-> ch ) )
4	3	ibi 176	. 2 |- ( ph -> ch )
5		pm5.21ni.2	. . . 4 |- ( ch -> ps )
6	5, 2	syl 14	. . 3 |- ( ch -> ( ph <-> ch ) )
7	6	ibir 177	. 2 |- ( ch -> ph )
8	4, 7	impbii 126	1 |- ( ph <-> ch )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anxordi  1444  elrabf  2960  sbcco  3053  sbc5  3055  sbcan  3074  sbcor  3076  sbcal  3083  sbcex2  3085  sbcel1v  3094  eldif  3209  elun  3348  elin  3390  elif  3617  rabsnif  3738  eluni  3896  eliun  3974  elopab  4352  opelopabsb  4354  opeliunxp  4781  opeliunxp2  4870  elxp4  5224  elxp5  5225  fsn2  5821  isocnv2  5953  elxp6  6332  elxp7  6333  opeliunxp2f  6404  brtpos2  6417  tpostpos  6430  ecdmn0m  6746  elixpsn  6904  bren  6917  omniwomnimkv  7366  elinp  7694  recexprlemell  7842  recexprlemelu  7843  gt0srpr  7968  ltresr  8059  eluz2  9761  elfz2  10250  infssuzex  10494  rexanuz2  11569  even2n  12453  infpn2  13095  xpsfrnel2  13447  issubg  13778  isnsg  13807  issrg  13997  iscrng2  14047  isrim0  14194  issubrng  14232  issubrg  14254  rrgval  14295  islssm  14390  islidlm  14512  2idlval  14535  2idlelb  14538  istopon  14756  ishmeo  15047  ismet2  15097  edgval  15930  istrl  16255  isclwwlk  16264  clwwlkn0  16278  isclwwlkn  16283  clwwlknonmpo  16298  clwwlknon  16299  clwwlk0on0  16301  iseupth  16317