Theorem csbcomg 3160

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 2868	. 2 |- (A e. V -> A e. _V)
2		elex 2868	. 2 |- (B e. W -> B e. _V)
3		sbccom 3118	. . . . . 6 |- ( [.A / x ]. [.B / y ].z e. C <-> [.B / y ]. [.A / x ].z e. C)
4	3	a1i 10	. . . . 5 |- ((A e. _V /\ B e. _V) -> ( [.A / x ]. [.B / y ].z e. C <-> [.B / y ]. [.A / x ].z e. C))
5		sbcel2g 3158	. . . . . . 7 |- (B e. _V -> ( [.B / y ].z e. C <-> z e. [_B / y]_C))
6	5	sbcbidv 3101	. . . . . 6 |- (B e. _V -> ( [.A / x ]. [.B / y ].z e. C <-> [.A / x ].z e. [_B / y]_C))
7	6	adantl 452	. . . . 5 |- ((A e. _V /\ B e. _V) -> ( [.A / x ]. [.B / y ].z e. C <-> [.A / x ].z e. [_B / y]_C))
8		sbcel2g 3158	. . . . . . 7 |- (A e. _V -> ( [.A / x ].z e. C <-> z e. [_A / x]_C))
9	8	sbcbidv 3101	. . . . . 6 |- (A e. _V -> ( [.B / y ]. [.A / x ].z e. C <-> [.B / y ].z e. [_A / x]_C))
10	9	adantr 451	. . . . 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 274	. . . 4 |- ((A e. _V /\ B e. _V) -> ( [.A / x ].z e. [_B / y]_C <-> [.B / y ].z e. [_A / x]_C))
12		sbcel2g 3158	. . . . 5 |- (A e. _V -> ( [.A / x ].z e. [_B / y]_C <-> z e. [_A / x]_[_B / y]_C))
13	12	adantr 451	. . . 4 |- ((A e. _V /\ B e. _V) -> ( [.A / x ].z e. [_B / y]_C <-> z e. [_A / x]_[_B / y]_C))
14		sbcel2g 3158	. . . . 5 |- (B e. _V -> ( [.B / y ].z e. [_A / x]_C <-> z e. [_B / y]_[_A / x]_C))
15	14	adantl 452	. . . 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 274	. . 3 |- ((A e. _V /\ B e. _V) -> (z e. [_A / x]_[_B / y]_C <-> z e. [_B / y]_[_A / x]_C))
17	16	eqrdv 2351	. 2 |- ((A e. _V /\ B e. _V) -> [_A / x]_[_B / y]_C = [_B / y]_[_A / x]_C)
18	1, 2, 17	syl2an 463	1 |- ((A e. V /\ B e. W) -> [_A / x]_[_B / y]_C = [_B / y]_[_A / x]_C)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)