Theorem r19.41vv

Description: Version of r19.41v with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017)

	Ref	Expression
Assertion	r19.41vv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.41v	⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) )
2	1	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) )
3		r19.41v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ 𝜓 ) )