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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