Theorem sylg 1823

Description: A syllogism combined with generalization. Inference associated with sylgt 1822. General form of alrimih 1824. (Contributed by BJ, 4-Oct-2019.)

Hypotheses

Ref	Expression
sylg.1	⊢ (𝜑 → ∀𝑥𝜓)
sylg.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
sylg	⊢ (𝜑 → ∀𝑥𝜒)

Proof of Theorem sylg

Step	Hyp	Ref	Expression
1		sylg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
2		sylg.2	. . 3 ⊢ (𝜓 → 𝜒)
3	2	alimi 1812	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1796  ax-4 1810

This theorem is referenced by:  alrimih  1824  ax9ALT  2819  ssrel  5659  kmlem1  9578  bnj1476  32121  bnj1533  32126  bj-alrimd  33955  bj-exlimd  33960  bj-ax12ig  33971  axc11n11  34018  bj-modalbe  34024  bj-modal4  34050  bj-wnfanf  34055  bj-wnfenf  34056  bj-19.12  34092  mpobi123f  35442  mptbi12f  35446