Theorem ala1 1846

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 1844	1 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-gen 1828  ax-4 1842

This theorem is used by:  19.38  1872  stdpc4  2105  ax12dgen  2172  ax12  2457  sb4a  2514  alral  3096  hbimtg  36309  mh-regprimbi  37089  bj-axdd2  37218  bj-ala1i  37244  bj-ax12v3ALT  37344  bj-equsal1t  37490