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Theorem wl-alanbii 34967
Description: This theorem extends alanimi 1818 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
Hypothesis
Ref Expression
wl-alanbii.1 (𝜑 ↔ (𝜓𝜒))
Assertion
Ref Expression
wl-alanbii (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Proof of Theorem wl-alanbii
StepHypRef Expression
1 wl-alanbii.1 . . 3 (𝜑 ↔ (𝜓𝜒))
21albii 1821 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝜓𝜒))
3 19.26 1871 . 2 (∀𝑥(𝜓𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
42, 3bitri 278 1 (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator