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Theorem hbn1w 1971
Description: Weak version of hbn1 2018. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
hbn1w (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem hbn1w
StepHypRef Expression
1 ax-5 1837 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 ax-5 1837 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
3 ax-5 1837 . 2 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
4 ax-5 1837 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
5 ax-5 1837 . 2 (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)
6 hbn1w.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
71, 2, 3, 4, 5, 6hbn1fw 1970 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  hba1w  1972  hba1wOLD  1973  hbe1w  1974  ax10w  2004  wl-naevhba1v  33275
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