Theorem pm5.21nii 709

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  1442  elrabf  2957  sbcco  3050  sbc5  3052  sbcan  3071  sbcor  3073  sbcal  3080  sbcex2  3082  sbcel1v  3091  eldif  3206  elun  3345  elin  3387  elif  3614  eluni  3891  eliun  3969  elopab  4346  opelopabsb  4348  opeliunxp  4774  opeliunxp2  4862  elxp4  5216  elxp5  5217  fsn2  5811  isocnv2  5942  elxp6  6321  elxp7  6322  opeliunxp2f  6390  brtpos2  6403  tpostpos  6416  ecdmn0m  6732  elixpsn  6890  bren  6903  omniwomnimkv  7345  elinp  7672  recexprlemell  7820  recexprlemelu  7821  gt0srpr  7946  ltresr  8037  eluz2  9739  elfz2  10223  infssuzex  10465  rexanuz2  11517  even2n  12400  infpn2  13042  xpsfrnel2  13394  issubg  13725  isnsg  13754  issrg  13943  iscrng2  13993  isrim0  14140  issubrng  14178  issubrg  14200  rrgval  14241  islssm  14336  islidlm  14458  2idlval  14481  2idlelb  14484  istopon  14702  ishmeo  14993  ismet2  15043  istrl  16124  isclwwlk  16132