Theorem axie2

Description: A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017)

	Ref	Expression
Assertion	axie2	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nf5	⊢ ( Ⅎ 𝑥 𝜓 ↔ ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) )
2		19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )
3	1 2	sylbir	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) ) )