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Theorem pm11.53v 1905
Description: Version of pm11.53 2178 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
Assertion
Ref Expression
pm11.53v (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem pm11.53v
StepHypRef Expression
1 19.21v 1867 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21albii 1746 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3 19.23v 1901 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
42, 3bitri 264 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887
This theorem depends on definitions:  df-bi 197  df-ex 1704
This theorem is referenced by:  sbnf2  2438
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