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Theorem bj-biexal3 36093
Description: When 𝜑 is substituted for 𝜓, both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019.)
Assertion
Ref Expression
bj-biexal3 (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))

Proof of Theorem bj-biexal3
StepHypRef Expression
1 bj-biexal1 36091 . 2 (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
2 bj-biexal2 36092 . 2 (∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))
31, 2bitr4i 278 1 (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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