Theorem sbccomg 2526

Description: Commutative law for double class substitution. (The proof was shortened by Eric Schmidt, 17-Jan-2007.)

Assertion

Ref	Expression
sbccomg	|- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

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

Proof of Theorem sbccomg

Step	Hyp	Ref	Expression
1		sbccog 2467	. . . 4 |- (A e. _V -> ([A / z][z / x][B / w][w / y]ph <-> [A / x][B / w][w / y]ph))
2	1	adantr 425	. . 3 |- ((A e. _V /\ B e. _V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / x][B / w][w / y]ph))
3		visset 2295	. . . . . . . . 9 |- z e. _V
4		sbccomglem 2525	. . . . . . . . 9 |- ((B e. _V /\ z e. _V) -> ([B / w][z / x][w / y]ph <-> [z / x][B / w][w / y]ph))
5	3, 4	mpan2 760	. . . . . . . 8 |- (B e. _V -> ([B / w][z / x][w / y]ph <-> [z / x][B / w][w / y]ph))
6		visset 2295	. . . . . . . . . 10 |- w e. _V
7		sbccomglem 2525	. . . . . . . . . 10 |- ((z e. _V /\ w e. _V) -> ([z / x][w / y]ph <-> [w / y][z / x]ph))
8	3, 6, 7	mp2an 761	. . . . . . . . 9 |- ([z / x][w / y]ph <-> [w / y][z / x]ph)
9	8	sbcbii 2506	. . . . . . . 8 |- (B e. _V -> ([B / w][z / x][w / y]ph <-> [B / w][w / y][z / x]ph))
10	5, 9	bitr3d 589	. . . . . . 7 |- (B e. _V -> ([z / x][B / w][w / y]ph <-> [B / w][w / y][z / x]ph))
11	10	sbcbidv 2505	. . . . . 6 |- ((B e. _V /\ A e. _V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / z][B / w][w / y][z / x]ph))
12	11	ancoms 484	. . . . 5 |- ((A e. _V /\ B e. _V) -> ([A / z][z / x][B / w][w / y]ph <-> [A / z][B / w][w / y][z / x]ph))
13		sbccomglem 2525	. . . . 5 |- ((A e. _V /\ B e. _V) -> ([A / z][B / w][w / y][z / x]ph <-> [B / w][A / z][w / y][z / x]ph))
14		sbccomglem 2525	. . . . . . 7 |- ((A e. _V /\ w e. _V) -> ([A / z][w / y][z / x]ph <-> [w / y][A / z][z / x]ph))
15	6, 14	mpan2 760	. . . . . 6 |- (A e. _V -> ([A / z][w / y][z / x]ph <-> [w / y][A / z][z / x]ph))
16	15	sbcbidv 2505	. . . . 5 |- ((A e. _V /\ B e. _V) -> ([B / w][A / z][w / y][z / x]ph <-> [B / w][w / y][A / z][z / x]ph))
17	12, 13, 16	3bitrd 603	. . . 4 |- ((A e. _V /\ B e. _V) -> ([A / z][z / x][B / w][w / y]ph <-> [B / w][w / y][A / z][z / x]ph))
18		sbccog 2467	. . . . 5 |- (B e. _V -> ([B / w][w / y][A / z][z / x]ph <-> [B / y][A / z][z / x]ph))
19	18	adantl 424	. . . 4 |- ((A e. _V /\ B e. _V) -> ([B / w][w / y][A / z][z / x]ph <-> [B / y][A / z][z / x]ph))
20		sbccog 2467	. . . . 5 |- (A e. _V -> ([A / z][z / x]ph <-> [A / x]ph))
21	20	sbcbidv 2505	. . . 4 |- ((A e. _V /\ B e. _V) -> ([B / y][A / z][z / x]ph <-> [B / y][A / x]ph))
22	17, 19, 21	3bitrd 603	. . 3 |- ((A e. _V /\ B e. _V) -> ([A / z][z / x][B / w][w / y]ph <-> [B / y][A / x]ph))
23		sbccog 2467	. . . . 5 |- (B e. _V -> ([B / w][w / y]ph <-> [B / y]ph))
24	23	sbcbidv 2505	. . . 4 |- ((B e. _V /\ A e. _V) -> ([A / x][B / w][w / y]ph <-> [A / x][B / y]ph))
25	24	ancoms 484	. . 3 |- ((A e. _V /\ B e. _V) -> ([A / x][B / w][w / y]ph <-> [A / x][B / y]ph))
26	2, 22, 25	3bitr3rd 608	. 2 |- ((A e. _V /\ B e. _V) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))
27		elisset 2299	. 2 |- (A e. C -> A e. _V)
28		elisset 2299	. 2 |- (B e. D -> B e. _V)
29	26, 27, 28	syl2an 503	1 |- ((A e. C /\ B e. D) -> ([A / x][B / y]ph <-> [B / y][A / x]ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  [wsbc 1534  _Vcvv 2292

This theorem is referenced by:  csbcomg 2560  csbabg 2588  csbabgOLD 2589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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