Theorem bibi1d 231

Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
imbid.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
bibi1d	|- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) )

Proof of Theorem bibi1d

Step	Hyp	Ref	Expression
1		imbid.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	bibi2d 230	. 2 |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) )
3		bicom 138	. 2 |- ( ( ps <-> th ) <-> ( th <-> ps ) )
4		bicom 138	. 2 |- ( ( ch <-> th ) <-> ( th <-> ch ) )
5	2, 3, 4	3bitr4g 221	1 |- ( ph -> ( ( ps <-> th ) <-> ( 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:  bibi12d  233  bibi1  238  biassdc  1327  eubidh  1948  eubid  1949  axext3  2065  bm1.1  2067  eqeq1  2088  pm13.183  2733  elabgt  2736  elrab3t  2749  mob  2775  sbctt  2881  sbcabel  2896  isoeq2  5473  caovcang  5693  frecabcl  6048  expap0  9603  bezoutlemeu  10540  dfgcd3  10543  bezout  10544  prmdvdsexp  10671  bdsepnft  10836  bdsepnfALT  10838  strcollnft  10937  strcollnfALT  10939