Theorem bj-cbvex2vv

Description: Version of cbvex2vv with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-cbval2vv.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-cbvex2vv	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbval2vv.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑧 𝜑
3		nfv	⊢ Ⅎ 𝑤 𝜑
4		nfv	⊢ Ⅎ 𝑥 𝜓
5		nfv	⊢ Ⅎ 𝑦 𝜓
6	2 3 4 5 1	cbvex2v	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜓 )