Theorem sbccom2f 33931

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

Hypotheses

Ref	Expression
sbccom2f.1	|- A e. _V
sbccom2f.2	|- F/_ y A

Assertion

Ref	Expression
sbccom2f	|- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

Distinct variable group:   x, y

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

Proof of Theorem sbccom2f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcco 3458	. . . 4 |- ( [. B / z ]. [. z / y ]. ph <-> [. B / y ]. ph )
2	1	bicomi 214	. . 3 |- ( [. B / y ]. ph <-> [. B / z ]. [. z / y ]. ph )
3	2	sbcbii 3491	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. [. B / z ]. [. z / y ]. ph )
4		sbccom2f.1	. . 3 |- A e. _V
5	4	sbccom2 33930	. 2 |- ( [. A / x ]. [. B / z ]. [. z / y ]. ph <-> [. [_ A / x ]_ B / z ]. [. A / x ]. [. z / y ]. ph )
6		vex 3203	. . . . . . 7 |- z e. _V
7	6	sbccom2 33930	. . . . . 6 |- ( [. z / y ]. [. A / x ]. ph <-> [. [_ z / y ]_ A / x ]. [. z / y ]. ph )
8		sbccom2f.2	. . . . . . . 8 |- F/_ y A
9		eqidd 2623	. . . . . . . 8 |- ( y = z -> A = A )
10	6, 8, 9	csbief 3558	. . . . . . 7 |- [_ z / y ]_ A = A
11		dfsbcq 3437	. . . . . . 7 |- ( [_ z / y ]_ A = A -> ( [. [_ z / y ]_ A / x ]. [. z / y ]. ph <-> [. A / x ]. [. z / y ]. ph ) )
12	10, 11	ax-mp 5	. . . . . 6 |- ( [. [_ z / y ]_ A / x ]. [. z / y ]. ph <-> [. A / x ]. [. z / y ]. ph )
13	7, 12	bitri 264	. . . . 5 |- ( [. z / y ]. [. A / x ]. ph <-> [. A / x ]. [. z / y ]. ph )
14	13	bicomi 214	. . . 4 |- ( [. A / x ]. [. z / y ]. ph <-> [. z / y ]. [. A / x ]. ph )
15	14	sbcbii 3491	. . 3 |- ( [. [_ A / x ]_ B / z ]. [. A / x ]. [. z / y ]. ph <-> [. [_ A / x ]_ B / z ]. [. z / y ]. [. A / x ]. ph )
16		sbcco 3458	. . 3 |- ( [. [_ A / x ]_ B / z ]. [. z / y ]. [. A / x ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )
17	15, 16	bitri 264	. 2 |- ( [. [_ A / x ]_ B / z ]. [. A / x ]. [. z / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )
18	3, 5, 17	3bitri 286	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _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-3an 1039  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:  sbccom2fi  33932