Theorem sylg 1820

Description: A syllogism combined with generalization. Inference associated with sylgt 1819. General form of alrimih 1821. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1821. (Revised 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 1808	. 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 1792  ax-4 1806

This theorem is referenced by:  alrimih  1821  ax9ALT  2730  raleqbidvvOLD  3333  csbied  3946  rzal  4515  ssrel  5795  ssrelOLD  5796  kmlem1  10189  bnj1476  34840  bnj1533  34845  bj-alrimd  36603  bj-exlimd  36608  bj-ax12ig  36619  axc11n11  36665  bj-modalbe  36671  bj-modal4  36697  bj-wnfanf  36702  bj-wnfenf  36703  bj-19.12  36744  bj-pm11.53vw  36759  mpobi123f  38149  mptbi12f  38153  ismnushort  44297  setrec2mpt  48928