Theorem cbvexv1

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex with a disjoint variable condition, which does not require ax-13 . See cbvexvw for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv for another variant. (Contributed by NM, 21-Jun-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbvalv1.nf1	⊢ Ⅎ 𝑦 𝜑
	cbvalv1.nf2	⊢ Ⅎ 𝑥 𝜓
	cbvalv1.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvexv1	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	⊢ Ⅎ 𝑦 𝜑
2		cbvalv1.nf2	⊢ Ⅎ 𝑥 𝜓
3		cbvalv1.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	nfn	⊢ Ⅎ 𝑦 ¬ 𝜑
5	2	nfn	⊢ Ⅎ 𝑥 ¬ 𝜓
6	3	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
7	4 5 6	cbvalv1	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ∀ 𝑦 ¬ 𝜓 )
8		alnex	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
9		alnex	⊢ ( ∀ 𝑦 ¬ 𝜓 ↔ ¬ ∃ 𝑦 𝜓 )
10	7 8 9	3bitr3i	⊢ ( ¬ ∃ 𝑥 𝜑 ↔ ¬ ∃ 𝑦 𝜓 )
11	10	con4bii	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )