Metamath Proof Explorer


Theorem bj-eeanvw

Description: Version of exdistrv with a disjoint variable condition on x , y not requiring ax-11 . (The same can be done with eeeanv and ee4anv .) (Contributed by BJ, 29-Sep-2019) (Proof modification is discouraged.)

Ref Expression
Assertion bj-eeanvw ( ∃ 𝑥𝑦 ( 𝜑𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.42v ( ∃ 𝑦 ( 𝜑𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2 1 exbii ( ∃ 𝑥𝑦 ( 𝜑𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
3 19.41v ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )
4 2 3 bitri ( ∃ 𝑥𝑦 ( 𝜑𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )