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Metamath Proof Explorer

Theorem mp2and

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004)

Ref Expression
Hypotheses mp2and.1 ⊢ ( 𝜑 → 𝜓 )
mp2and.2 ⊢ ( 𝜑 → 𝜒 )
mp2and.3 ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
Assertion mp2and ( 𝜑 → 𝜃 )

Proof

Step Hyp Ref Expression
1 mp2and.1 ⊢ ( 𝜑 → 𝜓 )
2 mp2and.2 ⊢ ( 𝜑 → 𝜒 )
3 mp2and.3 ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
4 1 3 mpand ⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
5 2 4 mpd ⊢ ( 𝜑 → 𝜃 )