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Metamath Proof Explorer

Theorem cbviunvg

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbviunv is preferred. (Contributed by NM, 15-Sep-2003) (New usage is discouraged.)

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