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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 𝜑
mpanl1.2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
Assertion
Ref Expression
mpanl1 ((𝜓𝜒) → 𝜃)

Proof of Theorem mpanl1
StepHypRef Expression
1 mpanl1.1 . . 3 𝜑
21jctl 314 . 2 (𝜓 → (𝜑𝜓))
3 mpanl1.2 . 2 (((𝜑𝜓) ∧ 𝜒) → 𝜃)
42, 3sylan 283 1 ((𝜓𝜒) → 𝜃)
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  6558  rec11api  8712  divdiv23apzi  8724  recp1lt1  8858  divgt0i  8869  divge0i  8870  ltreci  8871  lereci  8872  lt2msqi  8873  le2msqi  8874  msq11i  8875  ltdiv23i  8885  fnn0ind  9371  elfzp1b  10099  elfzm1b  10100  sqrt11i  11143  sqrtmuli  11144  sqrtmsq2i  11146  sqrtlei  11147  sqrtlti  11148
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