Theorem sbccom 3061

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 3060	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. A / z ]. [. w / y ]. [. z / x ]. ph )
2		sbccomlem 3060	. . . . . . 7 |- ( [. w / y ]. [. z / x ]. ph <-> [. z / x ]. [. w / y ]. ph )
3	2	sbcbii 3045	. . . . . 6 |- ( [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. B / w ]. [. z / x ]. [. w / y ]. ph )
4		sbccomlem 3060	. . . . . 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 3045	. . . 4 |- ( [. A / z ]. [. B / w ]. [. w / y ]. [. z / x ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
7		sbccomlem 3060	. . . . 5 |- ( [. A / z ]. [. w / y ]. [. z / x ]. ph <-> [. w / y ]. [. A / z ]. [. z / x ]. ph )
8	7	sbcbii 3045	. . . 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 3007	. . 3 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
11		sbcco 3007	. . 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 3007	. . 3 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
14	13	sbcbii 3045	. 2 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / y ]. ph )
15		sbcco 3007	. . 3 |- ( [. A / z ]. [. z / x ]. ph <-> [. A / x ]. ph )
16	15	sbcbii 3045	. 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 2985

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by:  csbcomg  3103  csbabg  3142  mpoxopovel  6294