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  2866  reu6i  2930  axsep2  4124  zfauscl  4125  copsexg  4246  euotd  4256  cnveq0  5087  iotaval  5191  iota5  5200  eufnfv  5749  isoeq1  5804  isoeq3  5806  isores2  5816  isores3  5818  isotr  5819  isoini2  5822  riota5f  5857  caovordg  6044  caovord  6048  dfoprab4f  6196  frecabcl  6402  nnaword  6514  xpf1o  6846  ltanqg  7401  ltmnqg  7402  ltasrg  7771  axpre-ltadd  7887  prmdvdsexp  12150  subrgsubm  13360  bdsep2  14723  bdzfauscl  14727