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	|- F/_ y B
	cbvmpt.2	|- F/_ x C
	cbvmpt.3	|- ( x = y -> B = C )
Assertion	cbvmpt	|- ( x e. A |-> B ) = ( y e. A |-> C )

Proof

Step	Hyp	Ref	Expression
1		cbvmpt.1	|- F/_ y B
2		cbvmpt.2	|- F/_ x C
3		cbvmpt.3	|- ( x = y -> B = C )
4		nfcv	|- F/_ x A
5		nfcv	|- F/_ y A
6	4 5 1 2 3	cbvmptf	|- ( x e. A |-> B ) = ( y e. A |-> C )