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Theorem bj-pm11.53vw 36110
Description: Version of pm11.53v 1939 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)
Assertion
Ref Expression
bj-pm11.53vw ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-pm11.53vw
StepHypRef Expression
1 simpl 482 . . . 4 ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → ∀𝑥Ⅎ'𝑦𝜑)
2 bj-19.21t 36103 . . . 4 (Ⅎ'𝑦𝜑 → (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
31, 2sylg 1817 . . 3 ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → ∀𝑥(∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓)))
4 albi 1812 . . 3 (∀𝑥(∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓)) → (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)))
53, 4syl 17 . 2 ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)))
6 bj-19.23t 36104 . . 3 (Ⅎ'𝑥𝑦𝜓 → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
76adantl 481 . 2 ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
85, 7bitrd 279 1 ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥𝑦𝜓) → (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531  wex 1773  Ⅎ'wnnf 36057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-bj-nnf 36058
This theorem is referenced by:  bj-pm11.53v  36111  bj-pm11.53a  36112
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