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Metamath Proof Explorer

Theorem cbv3

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3v if possible. (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv3.1	\|- F/ y ph
	cbv3.2	\|- F/ x ps
	cbv3.3	\|- ( x = y -> ( ph -> ps ) )
Assertion	cbv3	\|- ( A. x ph -> A. y ps )

Proof

Step	Hyp	Ref	Expression
1		cbv3.1	\|- F/ y ph
2		cbv3.2	\|- F/ x ps
3		cbv3.3	\|- ( x = y -> ( ph -> ps ) )
4	1	nf5ri	\|- ( ph -> A. y ph )
5	4	hbal	\|- ( A. x ph -> A. y A. x ph )
6	2 3	spim	\|- ( A. x ph -> ps )
7	5 6	alrimih	\|- ( A. x ph -> A. y ps )