Theorem sbccom 3338

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 3337	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph )
2		sbccomlem 3337	. . . . . . 7 |- ( [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. w / y ]. ph )
3	2	sbcbii 3322	. . . . . 6 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. z / x ]. [. w / y ]. ph )
4		sbccomlem 3337	. . . . . 6 |- ( [. B / w ]. [. z / x ]. [. w / y ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
5	3, 4	bitri 253	. . . . 5 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
6	5	sbcbii 3322	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
7		sbccomlem 3337	. . . . 5 |- ( [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. w / y ]. [. A / z ]. [. z / x ]. ph )
8	7	sbcbii 3322	. . . 4 |- ( [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
9	1, 6, 8	3bitr3i 279	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
10		sbcco 3289	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
11		sbcco 3289	. . 3 |- ( [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
12	9, 10, 11	3bitr3i 279	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
13		sbcco 3289	. . 3 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
14	13	sbcbii 3322	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / y ]. ph )
15		sbcco 3289	. . 3 |- ( [. A / z ]. [. z / x ]. ph <-> [. A / x ]. ph )
16	15	sbcbii 3322	. 2 |- ( [. B / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / x ]. ph )
17	12, 14, 16	3bitr3i 279	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 188   [.wsbc 3266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-sbc 3267

This theorem is referenced by:  csbcom  3782  csbab  3796  mpt2xopovel  6963  fi1uzind  12647  wrd2ind  12829  elmptrab  20835  sbccom2  32358  sbcrot3  35628  csbabgOLD  37205