Theorem cbveu

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbveuw , cbveuvw when possible. (Contributed by NM, 25-Nov-1994) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbveu.1	⊢ Ⅎ 𝑦 𝜑
	cbveu.2	⊢ Ⅎ 𝑥 𝜓
	cbveu.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbveu	⊢ ( ∃! 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	sb8eu	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5	2 3	sbie	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	5	eubii	⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃! 𝑦 𝜓 )
7	4 6	bitri	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )