Theorem cbvex2ivw

Description: One direction of cbvex2vw does not need a DV condition between x and y . Idea from relopabVD . (Contributed by SN, 27-Oct-2026)

	Ref	Expression
Hypothesis	cbvex2ivw.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 → 𝜓 ) )
Assertion	cbvex2ivw	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑧 ∃ 𝑤 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvex2ivw.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 → 𝜓 ) )
2		ax6evr	⊢ ∃ 𝑧 𝑥 = 𝑧
3		ax6evr	⊢ ∃ 𝑤 𝑦 = 𝑤
4	1	ex	⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑤 → ( 𝜑 → 𝜓 ) ) )
5	4	eximdv	⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑤 𝑦 = 𝑤 → ∃ 𝑤 ( 𝜑 → 𝜓 ) ) )
6	3 5	mpi	⊢ ( 𝑥 = 𝑧 → ∃ 𝑤 ( 𝜑 → 𝜓 ) )
7		19.37v	⊢ ( ∃ 𝑤 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑤 𝜓 ) )
8	6 7	sylib	⊢ ( 𝑥 = 𝑧 → ( 𝜑 → ∃ 𝑤 𝜓 ) )
9	2 8	eximii	⊢ ∃ 𝑧 ( 𝜑 → ∃ 𝑤 𝜓 )
10	9	19.37iv	⊢ ( 𝜑 → ∃ 𝑧 ∃ 𝑤 𝜓 )
11	10	exlimivv	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑧 ∃ 𝑤 𝜓 )