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Theorem pm11.53 1830
 Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.53 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem pm11.53
StepHypRef Expression
1 19.21v 1808 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21albii 1411 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3 ax-17 1471 . . . 4 (𝜓 → ∀𝑥𝜓)
43hbal 1418 . . 3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
5419.23h 1439 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
62, 5bitri 183 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1294  ∃wex 1433 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-i5r 1480 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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