Theorem cbvmowOLD

Description: Obsolete version of cbvmow as of 23-May-2024. (Contributed by NM, 9-Mar-1995) (Revised by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvmowOLD.1	⊢ Ⅎ 𝑦 𝜑
2		cbvmowOLD.2	⊢ Ⅎ 𝑥 𝜓
3		cbvmowOLD.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	sb8ev	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5	1	sb8euv	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
6	4 5	imbi12i	⊢ ( ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
7		moeu	⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) )
8		moeu	⊢ ( ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
9	6 7 8	3bitr4i	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
10	2 3	sbiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
11	10	mobii	⊢ ( ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃* 𝑦 𝜓 )
12	9 11	bitri	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )