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  1437  eubidh  2083  eubid  2084  axext3  2212  bm1.1  2214  eqeq1  2236  pm13.183  2941  elabgt  2944  elrab3t  2958  mob  2985  sbctt  3095  sbcabel  3111  isoeq2  5932  caovcang  6173  uchoice  6289  frecabcl  6551  expap0  10803  bezoutlemeu  12543  dfgcd3  12546  bezout  12547  prmdvdsexp  12685  ismet  15033  isxmet  15034  bdsepnft  16309  bdsepnfALT  16311