Theorem pm5.21nii 619

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 165	. 2 |- (ph -> ch)
5		pm5.21ni.2	. . . 4 |- (ch -> ps)
6	5, 2	syl 14	. . 3 |- (ch -> (ph <-> ch))
7	6	ibir 166	. 2 |- (ch -> ph)
8	4, 7	impbii 117	1 |- (ph <-> ch)

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anxordi  1288  elrabf  2690  sbcco  2779  sbc5  2781  sbcan  2799  sbcor  2801  sbcal  2804  sbcex2  2806  eldif  2921  elun  3078  elin  3120  eluni  3574  eliun  3652  elopab  3986  opelopabsb  3988  opeliunxp  4338  opeliunxp2  4419  elxp4  4751  elxp5  4752  fsn2  5280  isocnv2  5395  elxp6  5738  elxp7  5739  brtpos2  5807  tpostpos  5820  ecdmn0m  6084  bren  6164  elinp  6456  recexprlemell  6592  recexprlemelu  6593  gt0srpr  6636  ltresr  6696  eluz2  8215  elfz2  8611