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Metamath Proof Explorer

Theorem sbcg

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf . (Contributed by Alan Sare, 10-Nov-2012) Reduce axiom usage. (Revised by GG, 12-Oct-2024)

Ref Expression
Assertion sbcg 
|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Proof

Step Hyp Ref Expression
1 df-sbc 
 |-  ( [. A / x ]. ph <-> A e. { x | ph } )
2 dfclel 
 |-  ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
3 df-clab 
 |-  ( y e. { x | ph } <-> [ y / x ] ph )
4 sbv 
 |-  ( [ y / x ] ph <-> ph )
5 3 4 bitri 
 |-  ( y e. { x | ph } <-> ph )
6 5 anbi2i 
 |-  ( ( y = A /\ y e. { x | ph } ) <-> ( y = A /\ ph ) )
7 6 exbii 
 |-  ( E. y ( y = A /\ y e. { x | ph } ) <-> E. y ( y = A /\ ph ) )
8 1 2 7 3bitrri 
 |-  ( E. y ( y = A /\ ph ) <-> [. A / x ]. ph )
9 dfclel 
 |-  ( A e. V <-> E. y ( y = A /\ y e. V ) )
10 9 biimpi 
 |-  ( A e. V -> E. y ( y = A /\ y e. V ) )
11 simpr 
 |-  ( ( y = A /\ ph ) -> ph )
12 11 ax-gen 
 |-  A. y ( ( y = A /\ ph ) -> ph )
13 19.23v 
 |-  ( A. y ( ( y = A /\ ph ) -> ph ) <-> ( E. y ( y = A /\ ph ) -> ph ) )
14 13 biimpi 
 |-  ( A. y ( ( y = A /\ ph ) -> ph ) -> ( E. y ( y = A /\ ph ) -> ph ) )
15 12 14 mp1i 
 |-  ( E. y ( y = A /\ y e. V ) -> ( E. y ( y = A /\ ph ) -> ph ) )
16 2a1 
 |-  ( y = A -> ( y e. V -> ( ph -> y = A ) ) )
17 16 imp 
 |-  ( ( y = A /\ y e. V ) -> ( ph -> y = A ) )
18 17 ancrd 
 |-  ( ( y = A /\ y e. V ) -> ( ph -> ( y = A /\ ph ) ) )
19 18 eximi 
 |-  ( E. y ( y = A /\ y e. V ) -> E. y ( ph -> ( y = A /\ ph ) ) )
20 19.37imv 
 |-  ( E. y ( ph -> ( y = A /\ ph ) ) -> ( ph -> E. y ( y = A /\ ph ) ) )
21 19 20 syl 
 |-  ( E. y ( y = A /\ y e. V ) -> ( ph -> E. y ( y = A /\ ph ) ) )
22 15 21 impbid 
 |-  ( E. y ( y = A /\ y e. V ) -> ( E. y ( y = A /\ ph ) <-> ph ) )
23 10 22 syl 
 |-  ( A e. V -> ( E. y ( y = A /\ ph ) <-> ph ) )
24 8 23 bitr3id 
 |-  ( A e. V -> ( [. A / x ]. ph <-> ph ) )