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Theorem axc7e 2258
Description: Abbreviated version of axc7 2257 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
Assertion
Ref Expression
axc7e (∃𝑥𝑥𝜑𝜑)

Proof of Theorem axc7e
StepHypRef Expression
1 hbe1a 2082 . 2 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
2119.21bi 2117 1 (∃𝑥𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1505  wex 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-ex 1743
This theorem is referenced by:  19.9ht  2260  bj-axc10  33560  bj-19.12  33786
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