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Metamath Proof Explorer

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 )