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Theorem wl-nf-nf2 32463
 Description: By ax-10 2006 the definition nf5 2102 is stricter than the df-nf 1701 style. (Contributed by Wolf Lammen, 14-Sep-2021.)
Hypothesis
Ref Expression
wl-nf-nf2.1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Assertion
Ref Expression
wl-nf-nf2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))

Proof of Theorem wl-nf-nf2
StepHypRef Expression
1 exim 1751 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 df-ex 1696 . . 3 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 wl-nf-nf2.1 . . . 4 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
43con1i 143 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
52, 4sylbi 206 . 2 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
61, 5syl6 34 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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