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Metamath Proof Explorer

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 GG, 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 )