Theorem sylancb 646

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

Hypotheses

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

Assertion

Ref	Expression
sylancb	⊢ (φ → θ)

Proof of Theorem sylancb

Step	Hyp	Ref	Expression
1		sylancb.1	. . 3 ⊢ (φ ↔ ψ)
2		sylancb.2	. . 3 ⊢ (φ ↔ χ)
3		sylancb.3	. . 3 ⊢ ((ψ ∧ χ) → θ)
4	1, 2, 3	syl2anb 465	. 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)