Theorem sylg 1823

Description: A syllogism combined with generalization. Inference associated with sylgt 1822. General form of alrimih 1824. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1824. (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 1811	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1795  ax-4 1809

This theorem is referenced by:  alrimih  1824  ax9ALT  2730  raleqbidvvOLD  3314  csbied  3910  rzal  4484  ssrel  5761  ssrelOLD  5762  kmlem1  10165  bnj1476  34878  bnj1533  34883  bj-alrimd  36638  bj-exlimd  36643  bj-ax12ig  36654  axc11n11  36700  bj-modalbe  36706  bj-modal4  36732  bj-wnfanf  36737  bj-wnfenf  36738  bj-19.12  36779  bj-pm11.53vw  36794  mpobi123f  38186  mptbi12f  38190  ismnushort  44325  setrec2mpt  49561