Theorem biadan2 456

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 392	. 2 |- ( ph <-> ( ps /\ ph ) )
3		biadan2.2	. . 3 |- ( ps -> ( ph <-> ch ) )
4	3	pm5.32i 454	. 2 |- ( ( ps /\ ph ) <-> ( ps /\ ch ) )
5	2, 4	bitri 184	1 |- ( ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> 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:  elab4g  2888  elpwb  3587  ssdifsn  3722  brab2a  4681  brab2ga  4703  elovmpo  6074  eqop2  6181  elnnnn0  9221  elixx3g  9903  elfzo2  10152  1nprm  12116