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Theorem wl-nf2-nf 32464
 Description: hba1 2137 is sufficient to let a df-nf 1701 style definition be stricter than nf5 2102. (Contributed by Wolf Lammen, 14-Sep-2021.)
Hypothesis
Ref Expression
wl-nf2-nf.1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Assertion
Ref Expression
wl-nf2-nf ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))

Proof of Theorem wl-nf2-nf
StepHypRef Expression
1 wl-nf2-nf.1 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
21imim2i 16 . 2 ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝑥𝜑))
3 19.38 1757 . 2 ((∃𝑥𝜑 → ∀𝑥𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
42, 3syl 17 1 ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → 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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