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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	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )

Proof

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