Theorem sbccom2 33930

Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypothesis

Ref	Expression
sbccom2.1	|- A e. _V

Assertion

Ref	Expression
sbccom2	|- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

Distinct variable groups:   x, y   y, A

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

Proof of Theorem sbccom2

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

Step	Hyp	Ref	Expression
1		sbcco 3458	. . . . . . 7 |- ( [. B / w ]. [. w / y ]. ph <-> [. B / y ]. ph )
2	1	bicomi 214	. . . . . 6 |- ( [. B / y ]. ph <-> [. B / w ]. [. w / y ]. ph )
3	2	sbcbii 3491	. . . . 5 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
4		sbcco 3458	. . . . . 6 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / x ]. [. B / w ]. [. w / y ]. ph )
5	4	bicomi 214	. . . . 5 |- ( [. A / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph )
6		vex 3203	. . . . . . 7 |- z e. _V
7	6	sbccom2lem 33929	. . . . . 6 |- ( [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. [_ z / x ]_ B / w ]. [. z / x ]. [. w / y ]. ph )
8	7	sbcbii 3491	. . . . 5 |- ( [. A / z ]. [. z / x ]. [. B / w ]. [. w / y ]. ph <-> [. A / z ]. [. [_ z / x ]_ B / w ]. [. z / x ]. [. w / y ]. ph )
9	3, 5, 8	3bitri 286	. . . 4 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / z ]. [. [_ z / x ]_ B / w ]. [. z / x ]. [. w / y ]. ph )
10		sbccom2.1	. . . . 5 |- A e. _V
11	10	sbccom2lem 33929	. . . 4 |- ( [. A / z ]. [. [_ z / x ]_ B / w ]. [. z / x ]. [. w / y ]. ph <-> [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / z ]. [. z / x ]. [. w / y ]. ph )
12		sbcco 3458	. . . . 5 |- ( [. A / z ]. [. z / x ]. [. w / y ]. ph <-> [. A / x ]. [. w / y ]. ph )
13	12	sbcbii 3491	. . . 4 |- ( [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / z ]. [. z / x ]. [. w / y ]. ph <-> [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph )
14	9, 11, 13	3bitri 286	. . 3 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph )
15		csbco 3543	. . . 4 |- [_ A / z ]_ [_ z / x ]_ B = [_ A / x ]_ B
16		dfsbcq 3437	. . . 4 |- ( [_ A / z ]_ [_ z / x ]_ B = [_ A / x ]_ B -> ( [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph <-> [. [_ A / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph ) )
17	15, 16	ax-mp 5	. . 3 |- ( [. [_ A / z ]_ [_ z / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph <-> [. [_ A / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph )
18	14, 17	bitri 264	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph )
19		sbccom 3509	. . 3 |- ( [. A / x ]. [. w / y ]. ph <-> [. w / y ]. [. A / x ]. ph )
20	19	sbcbii 3491	. 2 |- ( [. [_ A / x ]_ B / w ]. [. A / x ]. [. w / y ]. ph <-> [. [_ A / x ]_ B / w ]. [. w / y ]. [. A / x ]. ph )
21		sbcco 3458	. 2 |- ( [. [_ A / x ]_ B / w ]. [. w / y ]. [. A / x ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )
22	18, 20, 21	3bitri 286	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  sbccom2f  33931