Theorem ala1 1845

Description: Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
ala1	⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))

Proof of Theorem ala1

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝜑 → (𝜓 → 𝜑))
2	1	alimi 1843	1 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-gen 1827  ax-4 1841

This theorem is used by:  19.38  1871  stdpc4  2104  ax12dgen  2170  ax12  2452  sb4a  2509  alral  3091  hbimtg  36662  mh-regprimbi  37427  bj-axdd2  37556  bj-ala1i  37582  bj-ax12v3ALT  37682  bj-equsal1t  37828