Theorem wl-lem-nexmo

Description: This theorem provides a basic working step in proving theorems about E* or E! . (Contributed by Wolf Lammen, 3-Oct-2019)

	Ref	Expression
Assertion	wl-lem-nexmo	⊢ ( ¬ ∃ 𝑥 𝜑 → ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) )

Proof

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