Theorem 3exbii

Description: Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995)

	Ref	Expression
Hypothesis	3exbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	3exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		3exbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	exbii	⊢ ( ∃ 𝑧 𝜑 ↔ ∃ 𝑧 𝜓 )
3	2	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 )