Theorem ala1 1843

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 1841	1 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1825  ax-4 1839

This theorem is referenced by:  19.38  1869  stdpc4  2102  ax12dgen  2169  ax12  2455  sb4a  2512  alral  3094  hbimtg  36277  mh-regprimbi  37037  bj-axdd2  37166  bj-ala1i  37192  bj-ax12v3ALT  37292  bj-equsal1t  37438