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 } ) |
Ref | Expression | ||
---|---|---|---|
Assertion | csbsngVD | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 | |
|
2 | sbceqg | |
|
3 | 1 2 | e1a | |
4 | csbconstg | |
|
5 | 1 4 | e1a | |
6 | eqeq1 | |
|
7 | 5 6 | e1a | |
8 | bibi1 | |
|
9 | 8 | biimprd | |
10 | 3 7 9 | e11 | |
11 | 10 | gen11 | |
12 | abbi | |
|
13 | 12 | biimpi | |
14 | 11 13 | e1a | |
15 | csbab | |
|
16 | 15 | a1i | |
17 | 16 | eqcomd | |
18 | 1 17 | e1a | |
19 | eqeq1 | |
|
20 | 19 | biimpcd | |
21 | 14 18 20 | e11 | |
22 | df-sn | |
|
23 | 22 | ax-gen | |
24 | csbeq2 | |
|
25 | 24 | a1i | |
26 | 1 23 25 | e10 | |
27 | eqeq2 | |
|
28 | 27 | biimpd | |
29 | 21 26 28 | e11 | |
30 | df-sn | |
|
31 | eqeq2 | |
|
32 | 31 | biimprcd | |
33 | 29 30 32 | e10 | |
34 | 33 | in1 | |