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Theorem hbe1a 2149
 Description: Dual statement of hbe1 2148. Modified version of axc7e 2339 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 1782 . 2 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2 hbn1 2147 . . 3 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
32con1i 149 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
41, 3sylbi 220 1 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2146 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  nf5-1  2150  axc7e  2339  nfeqf2  2397  bj-19.41al  34049  bj-sb56  34051  bj-modal4  34105  bj-wnf2  34109  bj-subst  34112  bj-nnfa1  34147
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