Theorem cbviinvg

Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbviinv is preferred. (Contributed by Jeff Hankins, 26-Aug-2009) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbviunvg.1	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
Assertion	cbviinvg	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		cbviunvg.1	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
2		nfcv	⊢ Ⅎ 𝑦 𝐵
3		nfcv	⊢ Ⅎ 𝑥 𝐶
4	2 3 1	cbviing	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶