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Metamath Proof Explorer

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 ) )