Theorem sbccom 3107

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 3106	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph )
2		sbccomlem 3106	. . . . . . 7 |- ( [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. w / y ]. ph )
3	2	sbcbii 3091	. . . . . 6 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. z / x ]. [. w / y ]. ph )
4		sbccomlem 3106	. . . . . 6 |- ( [. B / w ]. [. z / x ]. [. w / y ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
5	3, 4	bitri 184	. . . . 5 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. B / w ]. [. w / y ]. ph )
6	5	sbcbii 3091	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
7		sbccomlem 3106	. . . . 5 |- ( [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. w / y ]. [. A / z ]. [. z / x ]. ph )
8	7	sbcbii 3091	. . . 4 |- ( [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
9	1, 6, 8	3bitr3i 210	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph )
10		sbcco 3053	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
11		sbcco 3053	. . 3 |- ( [. B / w ]. [. w / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
12	9, 10, 11	3bitr3i 210	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. B / y ]. [. A / z ]. [. z / x ]. ph )
13		sbcco 3053	. . 3 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
14	13	sbcbii 3091	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / y ]. ph )
15		sbcco 3053	. . 3 |- ( [. A / z ]. [. z / x ]. ph <-> [. A / x ]. ph )
16	15	sbcbii 3091	. 2 |- ( [. B / y ]. [. A / z ]. [. z / x ]. ph <-> [. B / y ]. [. A / x ]. ph )
17	12, 14, 16	3bitr3i 210	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 105   [.wsbc 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  csbcomg  3150  csbabg  3189  mpoxopovel  6406  wrd2ind  11303