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Metamath Proof Explorer

Theorem spc3egv

Description: Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008) Avoid ax-10 and ax-11 . (Revised by GG, 20-Aug-2023) (Proof shortened by Wolf Lammen, 25-Aug-2023)

Ref Expression
Hypothesis spc3egv.1 
|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
Assertion spc3egv 
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )

Proof

Step Hyp Ref Expression
1 spc3egv.1 
 |-  ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
2 elex 
 |-  ( A e. V -> A e. _V )
3 elex 
 |-  ( B e. W -> B e. _V )
4 elex 
 |-  ( C e. X -> C e. _V )
5 simp1 
 |-  ( ( A e. _V /\ B e. _V /\ C e. _V ) -> A e. _V )
6 1 3coml 
 |-  ( ( y = B /\ z = C /\ x = A ) -> ( ph <-> ps ) )
7 6 3expa 
 |-  ( ( ( y = B /\ z = C ) /\ x = A ) -> ( ph <-> ps ) )
8 7 pm5.74da 
 |-  ( ( y = B /\ z = C ) -> ( ( x = A -> ph ) <-> ( x = A -> ps ) ) )
9 8 spc2egv 
 |-  ( ( B e. _V /\ C e. _V ) -> ( ( x = A -> ps ) -> E. y E. z ( x = A -> ph ) ) )
10 19.37v 
 |-  ( E. z ( x = A -> ph ) <-> ( x = A -> E. z ph ) )
11 10 exbii 
 |-  ( E. y E. z ( x = A -> ph ) <-> E. y ( x = A -> E. z ph ) )
12 19.37v 
 |-  ( E. y ( x = A -> E. z ph ) <-> ( x = A -> E. y E. z ph ) )
13 11 12 bitri 
 |-  ( E. y E. z ( x = A -> ph ) <-> ( x = A -> E. y E. z ph ) )
14 9 13 imbitrdi 
 |-  ( ( B e. _V /\ C e. _V ) -> ( ( x = A -> ps ) -> ( x = A -> E. y E. z ph ) ) )
15 14 pm2.86d 
 |-  ( ( B e. _V /\ C e. _V ) -> ( x = A -> ( ps -> E. y E. z ph ) ) )
16 15 3adant1 
 |-  ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( x = A -> ( ps -> E. y E. z ph ) ) )
17 16 imp 
 |-  ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ x = A ) -> ( ps -> E. y E. z ph ) )
18 5 17 spcimedv 
 |-  ( ( A e. _V /\ B e. _V /\ C e. _V ) -> ( ps -> E. x E. y E. z ph ) )
19 2 3 4 18 syl3an 
 |-  ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )