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Metamath Proof Explorer

Theorem mp3an12

Description: An inference based on modus ponens. (Contributed by NM, 13-Jul-2005)

Ref Expression
Hypotheses mp3an12.1 ⊢ 𝜑
mp3an12.2 ⊢ 𝜓
mp3an12.3 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion mp3an12 ( 𝜒 → 𝜃 )

Proof

Step Hyp Ref Expression
1 mp3an12.1 ⊢ 𝜑
2 mp3an12.2 ⊢ 𝜓
3 mp3an12.3 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
4 1 3 mp3an1 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
5 2 4 mpan ⊢ ( 𝜒 → 𝜃 )