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Theorem ax10w 1954
 Description: Weak version of ax-10 1966 from which we can prove any ax-10 1966 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
ax10w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
ax10w (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ax10w
StepHypRef Expression
1 ax10w.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
21hbn1w 1922 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194  ∀wal 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885 This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695 This theorem is referenced by: (None)
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