Theorem bibi1d 233

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 232	. 2 |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) )
3		bicom 140	. 2 |- ( ( ps <-> th ) <-> ( th <-> ps ) )
4		bicom 140	. 2 |- ( ( ch <-> th ) <-> ( th <-> ch ) )
5	2, 3, 4	3bitr4g 223	1 |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) )

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:  bibi12d  235  bibi1  240  biassdc  1444  eubidh  2092  eubid  2093  axext3  2221  bm1.1  2223  eqeq1  2245  pm13.183  2964  elabgt  2967  elrab3t  2981  mob  3008  sbctt  3118  sbcabel  3134  isoeq2  5998  caovcang  6241  uchoice  6361  frecabcl  6660  expap0  10984  bezoutlemeu  12762  dfgcd3  12765  bezout  12766  prmdvdsexp  12904  ismet  15368  isxmet  15369  bdsepnft  16827  bdsepnfALT  16829