Theorem 19.37vv

Description: Theorem *11.46 in WhiteheadRussell p. 164. Theorem 19.37 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.37vv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		19.37v	⊢ ( ∃ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∃ 𝑦 𝜑 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜓 → 𝜑 ) ↔ ∃ 𝑥 ( 𝜓 → ∃ 𝑦 𝜑 ) )
3		19.37v	⊢ ( ∃ 𝑥 ( 𝜓 → ∃ 𝑦 𝜑 ) ↔ ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜓 → 𝜑 ) ↔ ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )