# 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

### Proof

Step Hyp Ref Expression
1 idn1 ${⊢}\left({A}\in {V}{\to }{A}\in {V}\right)$
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 ${⊢}\left\{{B}\right\}=\left\{{y}|{y}={B}\right\}$
23 22 ax-gen ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left\{{B}\right\}=\left\{{y}|{y}={B}\right\}$
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