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Theorem mp3anr1 1583
Description: An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
Hypotheses
Ref Expression
mp3anr1.1 𝜓
mp3anr1.2 ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
Assertion
Ref Expression
mp3anr1 ((𝜑 ∧ (𝜒𝜃)) → 𝜏)

Proof of Theorem mp3anr1
StepHypRef Expression
1 mp3anr1.1 . . 3 𝜓
2 mp3anr1.2 . . . 4 ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
32ancoms 451 . . 3 (((𝜓𝜒𝜃) ∧ 𝜑) → 𝜏)
41, 3mp3anl1 1580 . 2 (((𝜒𝜃) ∧ 𝜑) → 𝜏)
54ancoms 451 1 ((𝜑 ∧ (𝜒𝜃)) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 386  df-3an 1110
This theorem is referenced by:  vc2OLD  27948  vc0  27954  vcm  27956  nvaddsub4  28037  nvpi  28047  nvge0  28053  ipval3  28089  ipidsq  28090  lnoadd  28138  lnosub  28139  dipsubdir  28228
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