Theorem sylancbr 363

Description: A syllogism inference combined with contraction.

Hypotheses

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

Assertion

Ref	Expression
sylancbr	⊢ (θ → χ)

Proof of Theorem sylancbr

Step	Hyp	Ref	Expression
1		sylancbr.1	. . 3 ⊢ ((φ ∧ ψ) → χ)
2		sylancbr.2	. . 3 ⊢ (φ ↔ θ)
3		sylancbr.3	. . 3 ⊢ (ψ ↔ θ)
4	1, 2, 3	syl2anbr 351	. 2 ⊢ ((θ ∧ θ) → χ)
5	4	anidms 332	1 ⊢ (θ → χ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  sucxpdom 3652

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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