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Metamath Proof Explorer

Theorem mp3and

Description: A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016)

Ref Expression
Hypotheses mp3and.1 ⊢ φ → ψ
mp3and.2 ⊢ φ → χ
mp3and.3 ⊢ φ → θ
mp3and.4 ⊢ φ → ψ ∧ χ ∧ θ → τ
Assertion mp3and ⊢ φ → τ

Proof

Step Hyp Ref Expression
1 mp3and.1 ⊢ φ → ψ
2 mp3and.2 ⊢ φ → χ
3 mp3and.3 ⊢ φ → θ
4 mp3and.4 ⊢ φ → ψ ∧ χ ∧ θ → τ
5 1 2 3 3jca ⊢ φ → ψ ∧ χ ∧ θ
6 5 4 mpd ⊢ φ → τ