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Theorem hbe1w 2046
Description: Weak version of hbe1 2138. See comments for ax10w 2124. 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 1772 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1w.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32notbid 319 . . 3 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
43hbn1w 2044 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
51, 4hbxfrbi 1816 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
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