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Theorem ax10w 2161
Description: Weak version of ax-10 2174 from which we can prove any ax-10 2174 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2130 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2130 instead. (New usage is discouraged.)
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 2130 1 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by: (None)
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