Theorem bibi2d 232

Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Hypothesis

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

Assertion

Ref	Expression
bibi2d	|- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) )

Proof of Theorem bibi2d

Step	Hyp	Ref	Expression
1		imbid.1	. . . . 5 |- ( ph -> ( ps <-> ch ) )
2	1	pm5.74i 180	. . . 4 |- ( ( ph -> ps ) <-> ( ph -> ch ) )
3	2	bibi2i 227	. . 3 |- ( ( ( ph -> th ) <-> ( ph -> ps ) ) <-> ( ( ph -> th ) <-> ( ph -> ch ) ) )
4		pm5.74 179	. . 3 |- ( ( ph -> ( th <-> ps ) ) <-> ( ( ph -> th ) <-> ( ph -> ps ) ) )
5		pm5.74 179	. . 3 |- ( ( ph -> ( th <-> ch ) ) <-> ( ( ph -> th ) <-> ( ph -> ch ) ) )
6	3, 4, 5	3bitr4i 212	. 2 |- ( ( ph -> ( th <-> ps ) ) <-> ( ph -> ( th <-> ch ) ) )
7	6	pm5.74ri 181	1 |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) )

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:  bibi1d  233  bibi12d  235  biantr  952  bimsc1  963  eujust  2028  euf  2031  ceqex  2864  reu6i  2928  axsep2  4121  zfauscl  4122  copsexg  4243  euotd  4253  cnveq0  5084  iotaval  5188  iota5  5197  eufnfv  5745  isoeq1  5799  isoeq3  5801  isores2  5811  isores3  5813  isotr  5814  isoini2  5817  riota5f  5852  caovordg  6039  caovord  6043  dfoprab4f  6191  frecabcl  6397  nnaword  6509  xpf1o  6841  ltanqg  7396  ltmnqg  7397  ltasrg  7766  axpre-ltadd  7882  prmdvdsexp  12140  subrgsubm  13293  bdsep2  14498  bdzfauscl  14502