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Theorem nexmoOLD 2549
 Description: Obsolete version of nexmo 2548 as of 16-Oct-2022. (Contributed by BJ, 30-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nexmoOLD (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmoOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 alnex 1744 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2 pm2.21 121 . . . . . . 7 𝜑 → (𝜑𝑥 = 𝑦))
32alrimiv 1886 . . . . . 6 𝜑 → ∀𝑦(𝜑𝑥 = 𝑦))
43alimi 1774 . . . . 5 (∀𝑥 ¬ 𝜑 → ∀𝑥𝑦(𝜑𝑥 = 𝑦))
5 alcom 2095 . . . . 5 (∀𝑥𝑦(𝜑𝑥 = 𝑦) ↔ ∀𝑦𝑥(𝜑𝑥 = 𝑦))
64, 5sylib 210 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑦𝑥(𝜑𝑥 = 𝑦))
7619.2d 1935 . . 3 (∀𝑥 ¬ 𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
8 df-mo 2547 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
97, 8sylibr 226 . 2 (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
101, 9sylbir 227 1 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505  ∃wex 1742  ∃*wmo 2545 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-11 2093 This theorem depends on definitions:  df-bi 199  df-ex 1743  df-mo 2547 This theorem is referenced by: (None)
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