Metamath Proof Explorer

Theorem csbsngVD

Description: Virtual deduction proof of csbsng . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbsng is csbsngVD without virtual deductions and was automatically derived from csbsngVD .

1:: |- (. A e. V ->. A e. V ).
2:1: |- (. A e. V ->. ( [. A / x ]. y = B <-> [_ A / x ]_ y = [_ A / x ]_ B ) ).
3:1: |- (. A e. V ->. [_ A / x ]_ y = y ).
4:3: |- (. A e. V ->. ( [_ A / x ]_ y = [_ A / x ]_ B <-> y = [_ A / x ]_ B ) ).
5:2,4: |- (. A e. V ->. ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) ).
6:5: |- (. A e. V ->. A. y ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) ).
7:6: |- (. A e. V ->. { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } ).
8:1: |- (. A e. V ->. { y | [. A / x ]. y = B } = [_ A / x ]_ { y | y = B } ).
9:7,8: |- (. A e. V ->. [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } ).
10:: |- { B } = { y | y = B }
11:10: |- A. x { B } = { y | y = B }
12:1,11: |- (. A e. V ->. [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } ).
13:9,12: |- (. A e. V ->. [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } ).
14:: |- { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
15:13,14: |- (. A e. V ->. [_ A / x ]_ { B } = { [_ A / x ]_ B } ).
qed:15: |- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
(Contributed by Alan Sare, 10-Nov-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion csbsngVD 
|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )

Proof

Step Hyp Ref Expression
1 idn1 
 |-  (. A e. V ->. A e. V ).
2 sbceqg 
 |-  ( A e. V -> ( [. A / x ]. y = B <-> [_ A / x ]_ y = [_ A / x ]_ B ) )
3 1 2 e1a 
 |-  (. A e. V ->. ( [. A / x ]. y = B <-> [_ A / x ]_ y = [_ A / x ]_ B ) ).
4 csbconstg 
 |-  ( A e. V -> [_ A / x ]_ y = y )
5 1 4 e1a 
 |-  (. A e. V ->. [_ A / x ]_ y = y ).
6 eqeq1 
 |-  ( [_ A / x ]_ y = y -> ( [_ A / x ]_ y = [_ A / x ]_ B <-> y = [_ A / x ]_ B ) )
7 5 6 e1a 
 |-  (. A e. V ->. ( [_ A / x ]_ y = [_ A / x ]_ B <-> y = [_ A / x ]_ B ) ).
8 bibi1 
 |-  ( ( [. A / x ]. y = B <-> [_ A / x ]_ y = [_ A / x ]_ B ) -> ( ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) <-> ( [_ A / x ]_ y = [_ A / x ]_ B <-> y = [_ A / x ]_ B ) ) )
9 8 biimprd 
 |-  ( ( [. A / x ]. y = B <-> [_ A / x ]_ y = [_ A / x ]_ B ) -> ( ( [_ A / x ]_ y = [_ A / x ]_ B <-> y = [_ A / x ]_ B ) -> ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) ) )
10 3 7 9 e11 
 |-  (. A e. V ->. ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) ).
11 10 gen11 
 |-  (. A e. V ->. A. y ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) ).
12 abbi 
 |-  ( A. y ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) <-> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
13 12 biimpi 
 |-  ( A. y ( [. A / x ]. y = B <-> y = [_ A / x ]_ B ) -> { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } )
14 11 13 e1a 
 |-  (. A e. V ->. { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } ).
15 csbab 
 |-  [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B }
16 15 a1i 
 |-  ( A e. V -> [_ A / x ]_ { y | y = B } = { y | [. A / x ]. y = B } )
17 16 eqcomd 
 |-  ( A e. V -> { y | [. A / x ]. y = B } = [_ A / x ]_ { y | y = B } )
18 1 17 e1a 
 |-  (. A e. V ->. { y | [. A / x ]. y = B } = [_ A / x ]_ { y | y = B } ).
19 eqeq1 
 |-  ( { y | [. A / x ]. y = B } = [_ A / x ]_ { y | y = B } -> ( { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } <-> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } ) )
20 19 biimpcd 
 |-  ( { y | [. A / x ]. y = B } = { y | y = [_ A / x ]_ B } -> ( { y | [. A / x ]. y = B } = [_ A / x ]_ { y | y = B } -> [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } ) )
21 14 18 20 e11 
 |-  (. A e. V ->. [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } ).
22 df-sn 
 |-  { B } = { y | y = B }
23 22 ax-gen 
 |-  A. x { B } = { y | y = B }
24 csbeq2 
 |-  ( A. x { B } = { y | y = B } -> [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } )
25 24 a1i 
 |-  ( A e. V -> ( A. x { B } = { y | y = B } -> [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } ) )
26 1 23 25 e10 
 |-  (. A e. V ->. [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } ).
27 eqeq2 
 |-  ( [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } -> ( [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } <-> [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } ) )
28 27 biimpd 
 |-  ( [_ A / x ]_ { y | y = B } = { y | y = [_ A / x ]_ B } -> ( [_ A / x ]_ { B } = [_ A / x ]_ { y | y = B } -> [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } ) )
29 21 26 28 e11 
 |-  (. A e. V ->. [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } ).
30 df-sn 
 |-  { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B }
31 eqeq2 
 |-  ( { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B } -> ( [_ A / x ]_ { B } = { [_ A / x ]_ B } <-> [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } ) )
32 31 biimprcd 
 |-  ( [_ A / x ]_ { B } = { y | y = [_ A / x ]_ B } -> ( { [_ A / x ]_ B } = { y | y = [_ A / x ]_ B } -> [_ A / x ]_ { B } = { [_ A / x ]_ B } ) )
33 29 30 32 e10 
 |-  (. A e. V ->. [_ A / x ]_ { B } = { [_ A / x ]_ B } ).
34 33 in1 
 |-  ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )