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Metamath Proof Explorer

Theorem cbvmptvg

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmptv for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 19-Feb-2013) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbvmptvg.1	$⊢ x = y \to B = C$
	Assertion	cbvmptvg	$⊢ (x \in A ⟼ B) = (y \in A ⟼ C)$

Proof

Step	Hyp	Ref	Expression
1		cbvmptvg.1	$⊢ x = y \to B = C$
2		nfcv	$⊢ \underline{Ⅎ} y B$
3		nfcv	$⊢ \underline{Ⅎ} x C$
4	2 3 1	cbvmptg	$⊢ (x \in A ⟼ B) = (y \in A ⟼ C)$