Metamath Proof Explorer


Theorem csbresgVD

Description: Virtual deduction proof of csbres . 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. csbres is csbresgVD without virtual deductions and was automatically derived from csbresgVD .

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

Ref Expression
Assertion csbresgVD A V A / x B C = A / x B A / x C

Proof

Step Hyp Ref Expression
1 idn1 A V A V
2 csbconstg A V A / x V = V
3 1 2 e1a A V A / x V = V
4 xpeq2 A / x V = V A / x C × A / x V = A / x C × V
5 3 4 e1a A V A / x C × A / x V = A / x C × V
6 csbxp A / x C × V = A / x C × A / x V
7 6 a1i A V A / x C × V = A / x C × A / x V
8 1 7 e1a A V A / x C × V = A / x C × A / x V
9 eqeq2 A / x C × A / x V = A / x C × V A / x C × V = A / x C × A / x V A / x C × V = A / x C × V
10 9 biimpd A / x C × A / x V = A / x C × V A / x C × V = A / x C × A / x V A / x C × V = A / x C × V
11 5 8 10 e11 A V A / x C × V = A / x C × V
12 ineq2 A / x C × V = A / x C × V A / x B A / x C × V = A / x B A / x C × V
13 11 12 e1a A V A / x B A / x C × V = A / x B A / x C × V
14 csbin A / x B C × V = A / x B A / x C × V
15 14 a1i A V A / x B C × V = A / x B A / x C × V
16 1 15 e1a A V A / x B C × V = A / x B A / x C × V
17 eqeq2 A / x B A / x C × V = A / x B A / x C × V A / x B C × V = A / x B A / x C × V A / x B C × V = A / x B A / x C × V
18 17 biimpd A / x B A / x C × V = A / x B A / x C × V A / x B C × V = A / x B A / x C × V A / x B C × V = A / x B A / x C × V
19 13 16 18 e11 A V A / x B C × V = A / x B A / x C × V
20 df-res B C = B C × V
21 20 ax-gen x B C = B C × V
22 csbeq2 x B C = B C × V A / x B C = A / x B C × V
23 22 a1i A V x B C = B C × V A / x B C = A / x B C × V
24 1 21 23 e10 A V A / x B C = A / x B C × V
25 eqeq2 A / x B C × V = A / x B A / x C × V A / x B C = A / x B C × V A / x B C = A / x B A / x C × V
26 25 biimpd A / x B C × V = A / x B A / x C × V A / x B C = A / x B C × V A / x B C = A / x B A / x C × V
27 19 24 26 e11 A V A / x B C = A / x B A / x C × V
28 df-res A / x B A / x C = A / x B A / x C × V
29 eqeq2 A / x B A / x C = A / x B A / x C × V A / x B C = A / x B A / x C A / x B C = A / x B A / x C × V
30 29 biimprcd A / x B C = A / x B A / x C × V A / x B A / x C = A / x B A / x C × V A / x B C = A / x B A / x C
31 27 28 30 e10 A V A / x B C = A / x B A / x C
32 31 in1 A V A / x B C = A / x B A / x C