Theorem cbveuwOLD

Description: Obsolete version of cbveuw as of 23-May-2024. (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbveuwOLD.1	⊢ Ⅎ 𝑦 𝜑
2		cbveuwOLD.2	⊢ Ⅎ 𝑥 𝜓
3		cbveuwOLD.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	sb8euv	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5	2 3	sbiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	5	eubii	⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃! 𝑦 𝜓 )
7	4 6	bitri	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 𝜓 )