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  9936  elfz2  10428  infssuzex  10676  rexanuz2  11771  even2n  12657  infpn2  13396  xpsfrnel2  13716  issubg  14025  isnsg  14054  mgpplusg  14271  mgpbas  14274  ringidval  14314  issrg  14318  iscrng2  14368  opprringb  14435  isrim0  14517  opprlring  14553  issubrng  14556  issubrg  14578  rrgval  14619  opprdrng  14669  islssm  14743  islidlm  14865  2idlval  14888  2idlelb  14891  asclfval  15070  istopon  15163  ishmeo  15454  ismet2  15504  edgval  16399  istrl  16724  isclwwlk  16733  clwwlkn0  16747  isclwwlkn  16752  clwwlknonmpo  16767  clwwlknon  16768  clwwlk0on0  16770  iseupth  16786