Theorem mpanr12 666

Description: An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)

Hypotheses

Ref	Expression
mpanr12.1	⊢ ψ
mpanr12.2	⊢ χ
mpanr12.3	⊢ ((φ ∧ (ψ ∧ χ)) → θ)

Assertion

Ref	Expression
mpanr12	⊢ (φ → θ)

Proof of Theorem mpanr12

Step	Hyp	Ref	Expression
1		mpanr12.2	. 2 ⊢ χ
2		mpanr12.1	. . 3 ⊢ ψ
3		mpanr12.3	. . 3 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
4	2, 3	mpanr1 664	. 2 ⊢ ((φ ∧ χ) → θ)
5	1, 4	mpan2 652	1 ⊢ (φ → θ)

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:  sbthlem1  6204