Theorem biadan2 453

Description: Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
biadan2.1	|- ( ph -> ps )
biadan2.2	|- ( ps -> ( ph <-> ch ) )

Assertion

Ref	Expression
biadan2	|- ( ph <-> ( ps /\ ch ) )

Proof of Theorem biadan2

Step	Hyp	Ref	Expression
1		biadan2.1	. . 3 |- ( ph -> ps )
2	1	pm4.71ri 390	. 2 |- ( ph <-> ( ps /\ ph ) )
3		biadan2.2	. . 3 |- ( ps -> ( ph <-> ch ) )
4	3	pm5.32i 451	. 2 |- ( ( ps /\ ph ) <-> ( ps /\ ch ) )
5	2, 4	bitri 183	1 |- ( ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  elab4g  2879  elpwb  3576  ssdifsn  3711  brab2a  4664  brab2ga  4686  elovmpo  6050  eqop2  6157  elnnnn0  9178  elixx3g  9858  elfzo2  10106  1nprm  12068