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Metamath Proof Explorer

Theorem sylg

Description: A syllogism combined with generalization. Inference associated with sylgt . General form of alrimih . (Contributed by NM, 9-Jan-1993) Extract from proof of alrimih . (Revised by BJ, 4-Oct-2019)

	Ref	Expression
Hypotheses	sylg.1	$⊢ φ \to \forall x ψ$
	sylg.2	$⊢ ψ \to χ$
Assertion	sylg	$⊢ φ \to \forall x χ$

Proof

Step	Hyp	Ref	Expression
1		sylg.1	$⊢ φ \to \forall x ψ$
2		sylg.2	$⊢ ψ \to χ$
3	2	alimi	$⊢ \forall x ψ \to \forall x χ$
4	1 3	syl	$⊢ φ \to \forall x χ$