Theorem cbvex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvexvw , cbvexv1 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbval.1	⊢ Ⅎ 𝑦 𝜑
2		cbval.2	⊢ Ⅎ 𝑥 𝜓
3		cbval.3	⊢ ( 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	cbval	⊢ ( ∀ 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 )