Theorem mp4an 654

Description: An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)

Hypotheses

Ref	Expression
mp4an.1	⊢ φ
mp4an.2	⊢ ψ
mp4an.3	⊢ χ
mp4an.4	⊢ θ
mp4an.5	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)

Assertion

Ref	Expression
mp4an	⊢ τ

Proof of Theorem mp4an

Step	Hyp	Ref	Expression
1		mp4an.1	. . 3 ⊢ φ
2		mp4an.2	. . 3 ⊢ ψ
3	1, 2	pm3.2i 441	. 2 ⊢ (φ ∧ ψ)
4		mp4an.3	. . 3 ⊢ χ
5		mp4an.4	. . 3 ⊢ θ
6	4, 5	pm3.2i 441	. 2 ⊢ (χ ∧ θ)
7		mp4an.5	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)
8	3, 6, 7	mp2an 653	1 ⊢ τ

Syntax hints:   → wi 4   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  tfin0c  4497  tc0c  6163  tcdi  6164  tc1c  6165