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Theorem mpanl1 434
Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
Hypotheses
Ref Expression
mpanl1.1  |-  ph
mpanl1.2  |-  ( ( ( ph  /\  ps )  /\  ch )  ->  th )
Assertion
Ref Expression
mpanl1  |-  ( ( ps  /\  ch )  ->  th )

Proof of Theorem mpanl1
StepHypRef Expression
1 mpanl1.1 . . 3  |-  ph
21jctl 314 . 2  |-  ( ps 
->  ( ph  /\  ps ) )
3 mpanl1.2 . 2  |-  ( ( ( ph  /\  ps )  /\  ch )  ->  th )
42, 3sylan 283 1  |-  ( ( ps  /\  ch )  ->  th )
Colors of variables: wff set class
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  ax-ia3 108
This theorem is referenced by:  mpanl12  436  ercnv  6550  rec11api  8696  divdiv23apzi  8708  recp1lt1  8842  divgt0i  8853  divge0i  8854  ltreci  8855  lereci  8856  lt2msqi  8857  le2msqi  8858  msq11i  8859  ltdiv23i  8869  fnn0ind  9355  elfzp1b  10080  elfzm1b  10081  sqrt11i  11122  sqrtmuli  11123  sqrtmsq2i  11125  sqrtlei  11126  sqrtlti  11127
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