Theorem eupickbi

Description: Theorem *14.26 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 27-Dec-2018)

	Ref	Expression
Assertion	eupickbi	⊢ ( ∃! 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eupicka	⊢ ( ( ∃! 𝑥 𝜑 ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
2	1	ex	⊢ ( ∃! 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
3		euex	⊢ ( ∃! 𝑥 𝜑 → ∃ 𝑥 𝜑 )
4		exintr	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
5	3 4	syl5com	⊢ ( ∃! 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
6	2 5	impbid	⊢ ( ∃! 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )