Theorem sylancbr 647

Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)

Hypotheses

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

Assertion

Ref	Expression
sylancbr	⊢ (φ → θ)

Proof of Theorem sylancbr

Step	Hyp	Ref	Expression
1		sylancbr.1	. . 3 ⊢ (ψ ↔ φ)
2		sylancbr.2	. . 3 ⊢ (χ ↔ φ)
3		sylancbr.3	. . 3 ⊢ ((ψ ∧ χ) → θ)
4	1, 2, 3	syl2anbr 466	. 2 ⊢ ((φ ∧ φ) → θ)
5	4	anidms 626	1 ⊢ (φ → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ 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: (None)