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  944  fiintim  7104  eqinfti  7198  finomni  7318  frecuzrdgrclt  10649  seq3coll  11077  swrdswrd  11253  swrdccatin1  11273  swrdccatin2  11277  fprodsplitsn  12160  nninfctlemfo  12577  unct  13029  isnzr2  14164  dvcnp2cntop  15389  fsumdvdsmul  15681  perfectlem2  15690  upgrwlkcompim  16108  wlkv0  16115