Theorem wl-lem-nexmo 37569

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(𝜑 → 𝑥 = 𝑧))

Proof of Theorem wl-lem-nexmo

Step	Hyp	Ref	Expression
1		alnex 1780	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 123	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑧))
3	2	alimi 1810	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧))
4	1, 3	sylbir 235	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by: (None)