Theorem pm10.52

Description: Theorem *10.52 in WhiteheadRussell p. 155. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.52	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.23v	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) )
2		pm5.5	⊢ ( ∃ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → 𝜓 ) ↔ 𝜓 ) )
3	1 2	syl5bb	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ 𝜓 ) )