Theorem simplbiim 391

Description: Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
simplbiim.1	|- ( ph <-> ( ps /\ ch ) )
simplbiim.2	|- ( ch -> th )

Assertion

Ref	Expression
simplbiim	|- ( ph -> th )

Proof of Theorem simplbiim

Step	Hyp	Ref	Expression
1		simplbiim.1	. 2 |- ( ph <-> ( ps /\ ch ) )
2		simplbiim.2	. . 3 |- ( ch -> th )
3	2	adantl 277	. 2 |- ( ( ps /\ ch ) -> th )
4	1, 3	sylbi 121	1 |- ( ph -> th )

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:  mpodifsnif  6181  ixpm  7012  finct  7456  apsscn  8975  zltaddlt1le  10410  pfxccatin12lem3  11504  oddnn02np1  12647  dvdsprmpweqnn  13115  sgrpass  13723  drnglring  14607  ausgrusgrben  16409