Theorem cbvmptg

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. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmpt for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by NM, 11-Sep-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvmptg.1	|- F/_ y B
	cbvmptg.2	|- F/_ x C
	cbvmptg.3	|- ( x = y -> B = C )
Assertion	cbvmptg	|- ( x e. A |-> B ) = ( y e. A |-> C )

Proof

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