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Theorem hbe1w 1974
 Description: Weak version of hbe1 2019. See comments for ax10w 2004. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
hbe1w (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem hbe1w
StepHypRef Expression
1 df-ex 1703 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1w.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32notbid 308 . . 3 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
43hbn1w 1971 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
51, 4hbxfrbi 1750 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1479  ∃wex 1702 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: (None)
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