Theorem mp3and 1280

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

Hypotheses

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

Assertion

Ref	Expression
mp3and	⊢ (φ → τ)

Proof of Theorem mp3and

Step	Hyp	Ref	Expression
1		mp3and.1	. . 3 ⊢ (φ → ψ)
2		mp3and.2	. . 3 ⊢ (φ → χ)
3		mp3and.3	. . 3 ⊢ (φ → θ)
4	1, 2, 3	3jca 1132	. 2 ⊢ (φ → (ψ ∧ χ ∧ θ))
5		mp3and.4	. 2 ⊢ (φ → ((ψ ∧ χ ∧ θ) → τ))
6	4, 5	mpd 14	1 ⊢ (φ → τ)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by: (None)