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Theorem wl-lem-nexmo 34965
Description: This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
Assertion
Ref Expression
wl-lem-nexmo (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))

Proof of Theorem wl-lem-nexmo
StepHypRef Expression
1 alnex 1783 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 123 . . 3 𝜑 → (𝜑𝑥 = 𝑧))
32alimi 1813 . 2 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))
41, 3sylbir 238 1 (¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))
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-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by: (None)
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