Theorem nexmo1 35512

Description: If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.)

Assertion

Ref	Expression
nexmo1	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmo1

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		moeu 2667	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 236	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1779  ∃*wmo 2619  ∃!weu 2652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-mo 2621  df-eu 2653

This theorem is referenced by: (None)