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Theorem mpanr2 414
Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
Hypotheses
Ref Expression
mpanr2.1 𝜒
mpanr2.2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
Assertion
Ref Expression
mpanr2 ((𝜑𝜓) → 𝜃)

Proof of Theorem mpanr2
StepHypRef Expression
1 mpanr2.1 . . 3 𝜒
21jctr 298 . 2 (𝜓 → (𝜓𝜒))
3 mpanr2.2 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
42, 3sylan2 270 1 ((𝜑𝜓) → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem is referenced by:  op1steq  5805  prarloclemarch2  6515  prarloclemlt  6589  prsradd  6868  muleqadd  7647  divdivap1  7697  addltmul  8159  elfzp1b  8957  elfzm1b  8958  expp1zap  9301  expm1ap  9302
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