Theorem sb8eulem

Description: Lemma. Factor out the common proof skeleton of sb8euv and sb8eu . Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 24-Aug-2019) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023)

	Ref	Expression
Hypothesis	sb8eulem.nfsb	|- F/ y [ w / x ] ph
Assertion	sb8eulem	|- ( E! x ph <-> E! y [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sb8eulem.nfsb	|- F/ y [ w / x ] ph
2		sb8v	|- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		equsb3	|- ( [ w / x ] x = z <-> w = z )
4	3	sblbis	|- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> w = z ) )
5	4	albii	|- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. w ( [ w / x ] ph <-> w = z ) )
6		nfv	|- F/ y w = z
7	1 6	nfbi	|- F/ y ( [ w / x ] ph <-> w = z )
8		nfv	|- F/ w ( [ y / x ] ph <-> y = z )
9		sbequ	|- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
10		equequ1	|- ( w = y -> ( w = z <-> y = z ) )
11	9 10	bibi12d	|- ( w = y -> ( ( [ w / x ] ph <-> w = z ) <-> ( [ y / x ] ph <-> y = z ) ) )
12	7 8 11	cbvalv1	|- ( A. w ( [ w / x ] ph <-> w = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
13	2 5 12	3bitri	|- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
14	13	exbii	|- ( E. z A. x ( ph <-> x = z ) <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
15		eu6	|- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
16		eu6	|- ( E! y [ y / x ] ph <-> E. z A. y ( [ y / x ] ph <-> y = z ) )
17	14 15 16	3bitr4i	|- ( E! x ph <-> E! y [ y / x ] ph )