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Theorem bj-modal4e 32680
Description: Dual statement of hba1 2149 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)
Assertion
Ref Expression
bj-modal4e (∃𝑥𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem bj-modal4e
StepHypRef Expression
1 hba1 2149 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
2 alnex 1704 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3 2exnaln 1754 . . . 4 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥𝑥 ¬ 𝜑)
43con2bii 347 . . 3 (∀𝑥𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝑥𝜑)
51, 2, 43imtr3i 280 . 2 (¬ ∃𝑥𝜑 → ¬ ∃𝑥𝑥𝜑)
65con4i 113 1 (∃𝑥𝑥𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708
This theorem is referenced by: (None)
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