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  951  mapsnd  6960  fiintim  7228  eqinfti  7350  finomni  7470  frecuzrdgrclt  10830  seq3coll  11272  swrdswrd  11455  swrdccatin1  11475  swrdccatin2  11479  fprodsplitsn  12378  nninfctlemfo  12795  unct  13311  isnzr2  14464  dvcnp2cntop  15723  fsumdvdsmul  16019  perfectlem2  16028  upgrwlkcompim  16517  wlkv0  16524  trlsegvdeglem1  16615