Theorem sylg 1826

Description: A syllogism combined with generalization. Inference associated with sylgt 1825. General form of alrimih 1827. (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 1815	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1799  ax-4 1813

This theorem is referenced by:  alrimih  1827  ax9ALT  2733  raleqbidvv  3329  csbied  3866  rzal  4436  ssrel  5683  kmlem1  9837  bnj1476  32727  bnj1533  32732  bj-alrimd  34728  bj-exlimd  34733  bj-ax12ig  34744  axc11n11  34791  bj-modalbe  34797  bj-modal4  34823  bj-wnfanf  34828  bj-wnfenf  34829  bj-19.12  34870  bj-pm11.53vw  34885  mpobi123f  36247  mptbi12f  36251  ismnushort  41808