Theorem mpancom 650

Description: An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

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

Assertion

Ref	Expression
mpancom	⊢ (ψ → χ)

Proof of Theorem mpancom

Step	Hyp	Ref	Expression
1		mpancom.1	. 2 ⊢ (ψ → φ)
2		id 19	. 2 ⊢ (ψ → ψ)
3		mpancom.2	. 2 ⊢ ((φ ∧ ψ) → χ)
4	1, 2, 3	syl2anc 642	1 ⊢ (ψ → χ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

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

This theorem is used by:  mpan  651  spesbc  3128  nnc3n3p1  6279  nnc3n3p2  6280  nnc3p1n3p2  6281