Theorem cbvmpt

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011) Add disjoint variable condition to avoid ax-13 . See cbvmptg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024)

	Ref	Expression
Hypotheses	cbvmpt.1	⊢ Ⅎ 𝑦 𝐵
	cbvmpt.2	⊢ Ⅎ 𝑥 𝐶
	cbvmpt.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
Assertion	cbvmpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		cbvmpt.1	⊢ Ⅎ 𝑦 𝐵
2		cbvmpt.2	⊢ Ⅎ 𝑥 𝐶
3		cbvmpt.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
4		nfcv	⊢ Ⅎ 𝑥 𝐴
5		nfcv	⊢ Ⅎ 𝑦 𝐴
6	4 5 1 2 3	cbvmptf	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 )