Theorem sylancom 648

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)

Hypotheses

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

Assertion

Ref	Expression
sylancom	⊢ ((φ ∧ ψ) → θ)

Proof of Theorem sylancom

Step	Hyp	Ref	Expression
1		sylancom.1	. 2 ⊢ ((φ ∧ ψ) → χ)
2		simpr 447	. 2 ⊢ ((φ ∧ ψ) → ψ)
3		sylancom.2	. 2 ⊢ ((χ ∧ ψ) → θ)
4	1, 2, 3	syl2anc 642	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:  fimacnvdisj  5244