Theorem sylg 1866

Description: A syllogism combined with generalization. Inference associated with sylgt 1865. General form of alrimih 1867. (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 1855	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1839  ax-4 1853

This theorem is referenced by:  alrimih  1867  aev2  2105  trintOLD  5004  ssrel  5455  kmlem1  9307  bnj1476  31516  bnj1533  31521  bj-ax12ig  33194  axc11n11  33261  bj-modalbe  33267  bj-ax9-2  33462