Theorem sbccom 2889

Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Assertion

Ref	Expression
sbccom	|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

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

Allowed substitution hints:   ph( x, y)   A( x)   B( y)

Proof of Theorem sbccom

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccomlem 2888	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph )
2		sbccomlem 2888	. . . . . . 7 |- ( [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. w / y ]. ph )
3	2	sbcbii 2873	. . . . . 6 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. z / x ]. [. w / y ]. ph )
4		sbccomlem 2888	. . . . . 6 |- ( [. B / w ]. [. z / x ]. [. w / y ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
5	3, 4	bitri 182	. . . . 5 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
6	5	sbcbii 2873	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
7		sbccomlem 2888	. . . . 5 |- ( [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. w / y ]. [. A / z ]. [. z / x ]. ph )
8	7	sbcbii 2873	. . . 4 |- ( [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
9	1, 6, 8	3bitr3i 208	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
10		sbcco 2836	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
11		sbcco 2836	. . 3 |- ( [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
12	9, 10, 11	3bitr3i 208	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
13		sbcco 2836	. . 3 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
14	13	sbcbii 2873	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / y ]. ph )
15		sbcco 2836	. . 3 |- ( [. A / z ]. [. z / x ]. ph <-> [. A / x ]. ph )
16	15	sbcbii 2873	. 2 |- ( [. B / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / x ]. ph )
17	12, 14, 16	3bitr3i 208	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 103   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  csbcomg  2929  csbabg  2963  mpt2xopovel  5879