Theorem r19.45zv

Description: Restricted version of Theorem 19.45 of Margaris p. 90. (Contributed by NM, 27-May-1998)

	Ref	Expression
Assertion	r19.45zv	⊢ ( 𝐴 ≠ ∅ → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.43	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		r19.9rzv	⊢ ( 𝐴 ≠ ∅ → ( 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜑 ) )
3	2	orbi1d	⊢ ( 𝐴 ≠ ∅ → ( ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ) )
4	1 3	bitr4id	⊢ ( 𝐴 ≠ ∅ → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ) )