Theorem sylg 1821

Description: A syllogism combined with generalization. Inference associated with sylgt 1820. General form of alrimih 1822. (Contributed by NM, 9-Jan-1993.) Extract from proof of alrimih 1822. (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 1809	. 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 1793  ax-4 1807

This theorem is referenced by:  alrimih  1822  ax9ALT  2735  raleqbidvvOLD  3343  csbied  3959  rzal  4532  ssrel  5806  ssrelOLD  5807  kmlem1  10220  bnj1476  34823  bnj1533  34828  bj-alrimd  36586  bj-exlimd  36591  bj-ax12ig  36602  axc11n11  36648  bj-modalbe  36654  bj-modal4  36680  bj-wnfanf  36685  bj-wnfenf  36686  bj-19.12  36727  bj-pm11.53vw  36742  mpobi123f  38122  mptbi12f  38126  ismnushort  44270  setrec2mpt  48789