Theorem cbveuvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	cbveuvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbveuvw	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbveuvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	cbvexvw	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )
3	1	cbvmovw	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )
4	2 3	anbi12i	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
5		df-eu	⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) )
6		df-eu	⊢ ( ∃! 𝑦 𝜓 ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
7	4 5 6	3bitr4i	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )