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Theorem hbn1fw 1969
Description: Weak version of ax-10 2016 from which we can prove any ax-10 2016 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
Hypotheses
Ref Expression
hbn1fw.1 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
hbn1fw.2 𝜓 → ∀𝑥 ¬ 𝜓)
hbn1fw.3 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
hbn1fw.4 𝜑 → ∀𝑦 ¬ 𝜑)
hbn1fw.5 (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)
hbn1fw.6 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
hbn1fw (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem hbn1fw
StepHypRef Expression
1 hbn1fw.1 . . . 4 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 hbn1fw.2 . . . 4 𝜓 → ∀𝑥 ¬ 𝜓)
3 hbn1fw.3 . . . 4 (∀𝑦𝜓 → ∀𝑥𝑦𝜓)
4 hbn1fw.4 . . . 4 𝜑 → ∀𝑦 ¬ 𝜑)
5 hbn1fw.6 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvalw 1965 . . 3 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
76notbii 310 . 2 (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓)
8 hbn1fw.5 . 2 (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)
97, 8hbxfrbi 1749 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  hbn1w  1970
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