Theorem mpanr2 665

Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanr2.1	⊢ χ
mpanr2.2	⊢ ((φ ∧ (ψ ∧ χ)) → θ)

Assertion

Ref	Expression
mpanr2	⊢ ((φ ∧ ψ) → θ)

Proof of Theorem mpanr2

Step	Hyp	Ref	Expression
1		mpanr2.1	. . 3 ⊢ χ
2	1	jctr 526	. 2 ⊢ (ψ → (ψ ∧ χ))
3		mpanr2.2	. 2 ⊢ ((φ ∧ (ψ ∧ χ)) → θ)
4	2, 3	sylan2 460	1 ⊢ ((φ ∧ ψ) → 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:  eladdci  4400