ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpanr2 GIF version

Theorem mpanr2 435
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 313 . 2 (𝜓 → (𝜓𝜒))
3 mpanr2.2 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
42, 3sylan2 284 1 ((𝜑𝜓) → 𝜃)
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:  op1steq  6084  fpmg  6575  pmresg  6577  pm54.43  7062  prarloclemarch2  7250  prarloclemlt  7324  prsradd  7617  muleqadd  8452  divdivap1  8506  addltmul  8979  elfzp1b  9907  elfzm1b  9908  expp1zap  10372  expm1ap  10373  fiinbas  12253  opnneissb  12361  blssec  12644
  Copyright terms: Public domain W3C validator