Theorem biadan2 460

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 396	. 2 |- ( ph <-> ( ps /\ ph ) )
3		biadan2.2	. . 3 |- ( ps -> ( ph <-> ch ) )
4	3	pm5.32i 458	. 2 |- ( ( ps /\ ph ) <-> ( ps /\ ch ) )
5	2, 4	bitri 184	1 |- ( ph <-> ( ps /\ ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  elab4g  2975  elpwb  3699  ssdifsn  3842  brab2a  4828  brab2ga  4850  elovmpo  6288  eqop2  6412  elnnnn0  9606  elixx3g  10303  elfzo2  10557  1nprm  12892