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	$$⊢\underset{\_}{Ⅎ}yB$$
	cbvmptg.2	$$⊢\underset{\_}{Ⅎ}xC$$
	cbvmptg.3	$$⊢x=y\to B=C$$
Assertion	cbvmptg	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvmptg.1	$$⊢\underset{\_}{Ⅎ}yB$$
2		cbvmptg.2	$$⊢\underset{\_}{Ⅎ}xC$$
3		cbvmptg.3	$$⊢x=y\to B=C$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
6	4 5 1 2 3	cbvmptfg	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼C\right)$$