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Metamath Proof Explorer

Theorem sb8

Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8f . (Contributed by NM, 16-May-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Jim Kingdon, 15-Jan-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb8.1	\|- F/ y ph
	Assertion	sb8	\|- ( A. x ph <-> A. y [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sb8.1	\|- F/ y ph
2	1	nfs1	\|- F/ x [ y / x ] ph
3		sbequ12	\|- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1 2 3	cbval	\|- ( A. x ph <-> A. y [ y / x ] ph )