Theorem 19.42

Description: Theorem 19.42 of Margaris p. 90. See 19.42v for a version requiring fewer axioms. See exan for an immediate version. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypothesis	19.42.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.42	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.42.1	⊢ Ⅎ 𝑥 𝜑
2	1	19.41	⊢ ( ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) ↔ ( ∃ 𝑥 𝜓 ∧ 𝜑 ) )
3		exancom	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜑 ) )
4		ancom	⊢ ( ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜓 ∧ 𝜑 ) )
5	2 3 4	3bitr4i	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) )