Theorem biadan2 444

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 384	. 2 |- ( ph <-> ( ps /\ ph ) )
3		biadan2.2	. . 3 |- ( ps -> ( ph <-> ch ) )
4	3	pm5.32i 442	. 2 |- ( ( ps /\ ph ) <-> ( ps /\ ch ) )
5	2, 4	bitri 182	1 |- ( ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  elab4g  2764  elpwb  3439  ssdifsn  3568  brab2a  4491  brab2ga  4513  elovmpt2  5845  eqop2  5948  elnnnn0  8716  elixx3g  9319  elfzo2  9561  1nprm  11374