Theorem pm5.21ndd 653

Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)

Hypotheses

Ref	Expression
pm5.21ndd.1	|- ( ph -> ( ch -> ps ) )
pm5.21ndd.2	|- ( ph -> ( th -> ps ) )
pm5.21ndd.3	|- ( ph -> ( ps -> ( ch <-> th ) ) )

Assertion

Ref	Expression
pm5.21ndd	|- ( ph -> ( ch <-> th ) )

Proof of Theorem pm5.21ndd

Step	Hyp	Ref	Expression
1		pm5.21ndd.1	. . . 4 |- ( ph -> ( ch -> ps ) )
2		pm5.21ndd.3	. . . 4 |- ( ph -> ( ps -> ( ch <-> th ) ) )
3	1, 2	syld 44	. . 3 |- ( ph -> ( ch -> ( ch <-> th ) ) )
4	3	ibd 176	. 2 |- ( ph -> ( ch -> th ) )
5		pm5.21ndd.2	. . . . 5 |- ( ph -> ( th -> ps ) )
6	5, 2	syld 44	. . . 4 |- ( ph -> ( th -> ( ch <-> th ) ) )
7		bicom1 129	. . . 4 |- ( ( ch <-> th ) -> ( th <-> ch ) )
8	6, 7	syl6 33	. . 3 |- ( ph -> ( th -> ( th <-> ch ) ) )
9	8	ibd 176	. 2 |- ( ph -> ( th -> ch ) )
10	4, 9	impbid 127	1 |- ( ph -> ( ch <-> th ) )

Syntax hints:   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.21nd  858  sbcrext  2891  rmob  2906  epelg  4045  eqbrrdva  4523  relbrcnvg  4724  fmptco  5351  ovelrn  5669  brtpos2  5889  brdomg  6252  genpelvl  6702  genpelvu  6703  fzoval  9158  clim  10120