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Theorem mpanr1 433
Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
Hypotheses
Ref Expression
mpanr1.1  |-  ps
mpanr1.2  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  ->  th )
Assertion
Ref Expression
mpanr1  |-  ( (
ph  /\  ch )  ->  th )

Proof of Theorem mpanr1
StepHypRef Expression
1 mpanr1.1 . 2  |-  ps
2 mpanr1.2 . . 3  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  ->  th )
32anassrs 397 . 2  |-  ( ( ( ph  /\  ps )  /\  ch )  ->  th )
41, 3mpanl2 431 1  |-  ( (
ph  /\  ch )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem is referenced by:  mpanr12  435  axcnre  7657  rec11api  8481  divdiv23apzi  8493  recp1lt1  8625  divgt0i  8636  divge0i  8637  ltreci  8638  lereci  8639  lt2msqi  8640  le2msqi  8641  msq11i  8642  ltdiv23i  8652  ge0gtmnf  9574  sqrt11i  10872  sqrtmuli  10873  sqrtmsq2i  10875  sqrtlei  10876  sqrtlti  10877  cos01gt0  11396
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