Theorem mp3and 1330

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 1167	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		mp3and.4	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))
6	4, 5	mpd 13	1 ⊢ (𝜑 → 𝜏)

Syntax hints:   → wi 4   ∧ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  eqsuptid  6958  eqinftid  6982  updjud  7043  seq3f1olemstep  10432  suprzcl2dc  11884  bezoutlemsup  11938