Theorem csbcomg 3030

Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Assertion

Ref	Expression
csbcomg	|- ( ( A e. V /\ B e. W ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )

Distinct variable groups:   y, A   x, B   x, y

Allowed substitution hints:   A( x)   B( y)   C( x, y)   V( x, y)   W( x, y)

Proof of Theorem csbcomg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2700	. 2 |- ( A e. V -> A e. _V )
2		elex 2700	. 2 |- ( B e. W -> B e. _V )
3		sbccom 2988	. . . . . 6 |- ( [. A / x ]. [. B / y ]. z e. C <-> [. B / y ]. [. A / x ]. z e. C )
4	3	a1i 9	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. z e. C <-> [. B / y ]. [. A / x ]. z e. C ) )
5		sbcel2g 3028	. . . . . . 7 |- ( B e. _V -> ( [. B / y ]. z e. C <-> z e. [_ B / y ]_ C ) )
6	5	sbcbidv 2971	. . . . . 6 |- ( B e. _V -> ( [. A / x ]. [. B / y ]. z e. C <-> [. A / x ]. z e. [_ B / y ]_ C ) )
7	6	adantl 275	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. z e. C <-> [. A / x ]. z e. [_ B / y ]_ C ) )
8		sbcel2g 3028	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. z e. C <-> z e. [_ A / x ]_ C ) )
9	8	sbcbidv 2971	. . . . . 6 |- ( A e. _V -> ( [. B / y ]. [. A / x ]. z e. C <-> [. B / y ]. z e. [_ A / x ]_ C ) )
10	9	adantr 274	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( [. B / y ]. [. A / x ]. z e. C <-> [. B / y ]. z e. [_ A / x ]_ C ) )
11	4, 7, 10	3bitr3d 217	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. z e. [_ B / y ]_ C <-> [. B / y ]. z e. [_ A / x ]_ C ) )
12		sbcel2g 3028	. . . . 5 |- ( A e. _V -> ( [. A / x ]. z e. [_ B / y ]_ C <-> z e. [_ A / x ]_ [_ B / y ]_ C ) )
13	12	adantr 274	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. z e. [_ B / y ]_ C <-> z e. [_ A / x ]_ [_ B / y ]_ C ) )
14		sbcel2g 3028	. . . . 5 |- ( B e. _V -> ( [. B / y ]. z e. [_ A / x ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C ) )
15	14	adantl 275	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( [. B / y ]. z e. [_ A / x ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C ) )
16	11, 13, 15	3bitr3d 217	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( z e. [_ A / x ]_ [_ B / y ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C ) )
17	16	eqrdv 2138	. 2 |- ( ( A e. _V /\ B e. _V ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )
18	1, 2, 17	syl2an 287	1 |- ( ( A e. V /\ B e. W ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   _Vcvv 2689   [.wsbc 2913   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  ovmpos  5902