Theorem sb8e

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8ev . (Contributed by NM, 12-Aug-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	sb8e	|- ( E. x ph <-> E. 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	cbvex	|- ( E. x ph <-> E. y [ y / x ] ph )