Theorem sylancb 415

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 289	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜃)
5	4	anidms 395	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)