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  7508  elinp  7842  recexprlemell  7990  recexprlemelu  7991  gt0srpr  8116  ltresr  8207  eluz2  9937  elfz2  10429  infssuzex  10677  rexanuz2  11773  even2n  12660  infpn2  13399  xpsfrnel2  13720  issubg  14029  isnsg  14058  cntrval  14145  mgpplusg  14306  mgpbas  14309  ringidval  14349  issrg  14353  iscrng2  14403  opprringb  14470  isrim0  14552  opprlring  14588  issubrng  14591  issubrg  14613  rrgval  14654  opprdrng  14704  islssm  14778  islidlm  14900  2idlval  14923  2idlelb  14926  asclfval  15105  istopon  15205  ishmeo  15496  ismet2  15546  edgval  16467  istrl  16792  isclwwlk  16801  clwwlkn0  16815  isclwwlkn  16820  clwwlknonmpo  16835  clwwlknon  16836  clwwlk0on0  16838  iseupth  16854