Theorem sbccom 2833

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 2832	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph )
2		sbccomlem 2832	. . . . . . 7 |- ( [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. w / y ]. ph )
3	2	sbcbii 2818	. . . . . 6 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. z / x ]. [. w / y ]. ph )
4		sbccomlem 2832	. . . . . 6 |- ( [. B / w ]. [. z / x ]. [. w / y ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
5	3, 4	bitri 173	. . . . 5 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
6	5	sbcbii 2818	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
7		sbccomlem 2832	. . . . 5 |- ( [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. w / y ]. [. A / z ]. [. z / x ]. ph )
8	7	sbcbii 2818	. . . 4 |- ( [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
9	1, 6, 8	3bitr3i 199	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
10		sbcco 2785	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
11		sbcco 2785	. . 3 |- ( [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
12	9, 10, 11	3bitr3i 199	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
13		sbcco 2785	. . 3 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
14	13	sbcbii 2818	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / y ]. ph )
15		sbcco 2785	. . 3 |- ( [. A / z ]. [. z / x ]. ph <-> [. A / x ]. ph )
16	15	sbcbii 2818	. 2 |- ( [. B / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / x ]. ph )
17	12, 14, 16	3bitr3i 199	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 98   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  csbcomg  2873  csbabg  2907  mpt2xopovel  5856