Theorem cbveuALT

Description: Alternative proof of cbveu . Since df-eu combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbveu.1	⊢ Ⅎ 𝑦 𝜑
2		cbveu.2	⊢ Ⅎ 𝑥 𝜓
3		cbveu.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1 2 3	cbvex	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )
5	1 2 3	cbvmo	⊢ ( ∃* 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 )