Theorem sylg 1825

Description: A syllogism combined with generalization. Inference associated with sylgt 1824. General form of alrimih 1826. (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 1813	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1797  ax-4 1811

This theorem is referenced by:  alrimih  1826  ax9ALT  2727  raleqbidvvOLD  3330  csbied  3930  rzal  4507  ssrel  5780  ssrelOLD  5781  kmlem1  10141  bnj1476  33846  bnj1533  33851  bj-alrimd  35485  bj-exlimd  35490  bj-ax12ig  35501  axc11n11  35548  bj-modalbe  35554  bj-modal4  35580  bj-wnfanf  35585  bj-wnfenf  35586  bj-19.12  35627  bj-pm11.53vw  35642  mpobi123f  37018  mptbi12f  37022  ismnushort  43045  setrec2mpt  47695