Theorem 19.42v

Description: Version of 19.42 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	19.42v	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.41v	⊢ ( ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) ↔ ( ∃ 𝑥 𝜓 ∧ 𝜑 ) )
2		exancom	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) )
3		ancom	⊢ ( ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜓 ∧ 𝜑 ) )
4	1 2 3	3bitr4i	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )