Theorem cbveuw

Description: Version of cbveu with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbveuw.1	⊢ Ⅎ 𝑦 𝜑
	cbveuw.2	⊢ Ⅎ 𝑥 𝜓
	cbveuw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbveuw	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbveuw.1	⊢ Ⅎ 𝑦 𝜑
2		cbveuw.2	⊢ Ⅎ 𝑥 𝜓
3		cbveuw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1 2 3	cbvexv1	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )
5	1 2 3	cbvmow	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )
6	4 5	anbi12i	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
7		df-eu	⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) )
8		df-eu	⊢ ( ∃! 𝑦 𝜓 ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
9	6 7 8	3bitr4i	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )