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Theorem pm11.53v 1962
 Description: Version of pm11.53 2215 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 1908 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21albii 1787 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3 19.23v 1911 . 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 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  sbnf2  2467
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