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  961  bimsc1  972  eujust  2081  euf  2084  ceqex  2934  reu6i  2998  axsep2  4213  zfauscl  4214  copsexg  4342  euotd  4353  cnveq0  5200  iotaval  5305  iota5  5315  eufnfv  5895  isoeq1  5952  isoeq3  5954  isores2  5964  isores3  5966  isotr  5967  isoini2  5970  riota5f  6008  caovordg  6200  caovord  6204  dfoprab4f  6365  frecabcl  6608  nnaword  6722  xpf1o  7073  ltanqg  7663  ltmnqg  7664  ltasrg  8033  axpre-ltadd  8149  prmdvdsexp  12783  subrgsubm  14312  wlkeq  16278  bdsep2  16585  bdzfauscl  16589