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Theorem hbe1a 2190
Description: Dual statement of hbe1 2189. Modified version of axc7e 2311 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
Assertion
Ref Expression
hbe1a (∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a
StepHypRef Expression
1 df-ex 1860 . 2 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2 hbn1 2188 . . 3 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
32con1i 146 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
41, 3sylbi 208 1 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1635  wex 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2187
This theorem depends on definitions:  df-bi 198  df-ex 1860
This theorem is referenced by:  nf5-1  2191  axc7e  2311  axc7eOLD  2312  nfeqf2  2466  bj-19.41al  32949  bj-sb56  32951
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