Theorem csbcomg 3117

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 2784	. 2 |- ( A e. V -> A e. _V )
2		elex 2784	. 2 |- ( B e. W -> B e. _V )
3		sbccom 3075	. . . . . 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 3115	. . . . . . 7 |- ( B e. _V -> ( [. B / y ]. z e. C <-> z e. [_ B / y ]_ C ) )
6	5	sbcbidv 3058	. . . . . 6 |- ( B e. _V -> ( [. A / x ]. [. B / y ]. z e. C <-> [. A / x ]. z e. [_ B / y ]_ C ) )
7	6	adantl 277	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. z e. C <-> [. A / x ]. z e. [_ B / y ]_ C ) )
8		sbcel2g 3115	. . . . . . 7 |- ( A e. _V -> ( [. A / x ]. z e. C <-> z e. [_ A / x ]_ C ) )
9	8	sbcbidv 3058	. . . . . 6 |- ( A e. _V -> ( [. B / y ]. [. A / x ]. z e. C <-> [. B / y ]. z e. [_ A / x ]_ C ) )
10	9	adantr 276	. . . . 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 218	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. z e. [_ B / y ]_ C <-> [. B / y ]. z e. [_ A / x ]_ C ) )
12		sbcel2g 3115	. . . . 5 |- ( A e. _V -> ( [. A / x ]. z e. [_ B / y ]_ C <-> z e. [_ A / x ]_ [_ B / y ]_ C ) )
13	12	adantr 276	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. z e. [_ B / y ]_ C <-> z e. [_ A / x ]_ [_ B / y ]_ C ) )
14		sbcel2g 3115	. . . . 5 |- ( B e. _V -> ( [. B / y ]. z e. [_ A / x ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C ) )
15	14	adantl 277	. . . 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 218	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( z e. [_ A / x ]_ [_ B / y ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C ) )
17	16	eqrdv 2204	. 2 |- ( ( A e. _V /\ B e. _V ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )
18	1, 2, 17	syl2an 289	1 |- ( ( A e. V /\ B e. W ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   _Vcvv 2773   [.wsbc 2999   [_csb 3094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3000  df-csb 3095

This theorem is referenced by:  ovmpos  6076