Theorem impel 280

Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)

Hypotheses

Ref	Expression
impel.1	|- ( ph -> ( ps -> ch ) )
impel.2	|- ( th -> ps )

Assertion

Ref	Expression
impel	|- ( ( ph /\ th ) -> ch )

Proof of Theorem impel

Step	Hyp	Ref	Expression
1		impel.2	. . 3 |- ( th -> ps )
2		impel.1	. . 3 |- ( ph -> ( ps -> ch ) )
3	1, 2	syl5 32	. 2 |- ( ph -> ( th -> ch ) )
4	3	imp 124	1 |- ( ( ph /\ th ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104

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

This theorem is referenced by:  pm4.55dc  947  fiintim  7166  eqinfti  7279  finomni  7399  frecuzrdgrclt  10740  seq3coll  11169  swrdswrd  11352  swrdccatin1  11372  swrdccatin2  11376  fprodsplitsn  12274  nninfctlemfo  12691  unct  13143  isnzr2  14279  dvcnp2cntop  15510  fsumdvdsmul  15805  perfectlem2  15814  upgrwlkcompim  16303  wlkv0  16310  trlsegvdeglem1  16401