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Metamath Proof Explorer

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 )