Theorem pm5.21nii 716

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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  anxordi  1449  elrabf  2980  sbcco  3073  sbc5  3075  sbcan  3094  sbcor  3096  sbcal  3103  sbcex2  3105  sbcel1v  3114  eldif  3229  elun  3370  elin  3412  elif  3652  rabsnif  3778  eluni  3938  eliun  4016  elopab  4400  opelopabsb  4402  opeliunxp  4830  opeliunxp2  4920  elxp4  5275  elxp5  5276  fsn2  5882  isocnv2  6018  elxp6  6403  elxp7  6404  opeliunxp2f  6509  brtpos2  6522  tpostpos  6535  ecdmn0m  6851  elixpsn  7017  bren  7030  omniwomnimkv  7507  elinp  7841  recexprlemell  7989  recexprlemelu  7990  gt0srpr  8115  ltresr  8206  eluz2  9927  elfz2  10418  infssuzex  10666  rexanuz2  11757  even2n  12641  infpn2  13347  xpsfrnel2  13667  issubg  13976  isnsg  14005  mgpplusg  14222  mgpbas  14225  ringidval  14265  issrg  14269  iscrng2  14319  opprringb  14386  isrim0  14468  opprlring  14504  issubrng  14507  issubrg  14529  rrgval  14570  opprdrng  14620  islssm  14694  islidlm  14816  2idlval  14839  2idlelb  14842  asclfval  15021  istopon  15114  ishmeo  15405  ismet2  15455  edgval  16301  istrl  16626  isclwwlk  16635  clwwlkn0  16649  isclwwlkn  16654  clwwlknonmpo  16669  clwwlknon  16670  clwwlk0on0  16672  iseupth  16688