Theorem wl-sbcom2d 33344

Description: Version of sbcom2 2445 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)

Hypotheses

Ref	Expression
wl-sbcom2d.1	|- ( ph -> -. A. x x = w )
wl-sbcom2d.2	|- ( ph -> -. A. x x = z )
wl-sbcom2d.3	|- ( ph -> -. A. z z = y )

Assertion

Ref	Expression
wl-sbcom2d	|- ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) )

Proof of Theorem wl-sbcom2d

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 |- E. u u = y
2		ax6ev 1890	. 2 |- E. v v = w
3		wl-sbcom2d.2	. . . . . . . 8 |- ( ph -> -. A. x x = z )
4		wl-sbcom2d-lem2 33343	. . . . . . . . . . 11 |- ( -. A. z z = x -> ( [ u / x ] [ v / z ] ps <-> A. x A. z ( ( x = u /\ z = v ) -> ps ) ) )
5		alcom 2037	. . . . . . . . . . . 12 |- ( A. x A. z ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( x = u /\ z = v ) -> ps ) )
6		ancomst 468	. . . . . . . . . . . . 13 |- ( ( ( x = u /\ z = v ) -> ps ) <-> ( ( z = v /\ x = u ) -> ps ) )
7	6	2albii 1748	. . . . . . . . . . . 12 |- ( A. z A. x ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) )
8	5, 7	bitri 264	. . . . . . . . . . 11 |- ( A. x A. z ( ( x = u /\ z = v ) -> ps ) <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) )
9	4, 8	syl6bb 276	. . . . . . . . . 10 |- ( -. A. z z = x -> ( [ u / x ] [ v / z ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) )
10	9	naecoms 2313	. . . . . . . . 9 |- ( -. A. x x = z -> ( [ u / x ] [ v / z ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) )
11		wl-sbcom2d-lem2 33343	. . . . . . . . 9 |- ( -. A. x x = z -> ( [ v / z ] [ u / x ] ps <-> A. z A. x ( ( z = v /\ x = u ) -> ps ) ) )
12	10, 11	bitr4d 271	. . . . . . . 8 |- ( -. A. x x = z -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) )
13	3, 12	syl 17	. . . . . . 7 |- ( ph -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) )
14	13	adantl 482	. . . . . 6 |- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ u / x ] [ v / z ] ps <-> [ v / z ] [ u / x ] ps ) )
15		wl-sbcom2d.1	. . . . . . . 8 |- ( ph -> -. A. x x = w )
16		wl-sbcom2d-lem1 33342	. . . . . . . 8 |- ( ( u = y /\ v = w ) -> ( -. A. x x = w -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) ) )
17	15, 16	syl5 34	. . . . . . 7 |- ( ( u = y /\ v = w ) -> ( ph -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) ) )
18	17	imp 445	. . . . . 6 |- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ u / x ] [ v / z ] ps <-> [ y / x ] [ w / z ] ps ) )
19		wl-sbcom2d.3	. . . . . . . . 9 |- ( ph -> -. A. z z = y )
20		wl-sbcom2d-lem1 33342	. . . . . . . . 9 |- ( ( v = w /\ u = y ) -> ( -. A. z z = y -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) )
21	19, 20	syl5 34	. . . . . . . 8 |- ( ( v = w /\ u = y ) -> ( ph -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) )
22	21	ancoms 469	. . . . . . 7 |- ( ( u = y /\ v = w ) -> ( ph -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) ) )
23	22	imp 445	. . . . . 6 |- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ v / z ] [ u / x ] ps <-> [ w / z ] [ y / x ] ps ) )
24	14, 18, 23	3bitr3rd 299	. . . . 5 |- ( ( ( u = y /\ v = w ) /\ ph ) -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) )
25	24	exp31 630	. . . 4 |- ( u = y -> ( v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) )
26	25	exlimdv 1861	. . 3 |- ( u = y -> ( E. v v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) )
27	26	exlimiv 1858	. 2 |- ( E. u u = y -> ( E. v v = w -> ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) ) ) )
28	1, 2, 27	mp2 9	1 |- ( ph -> ( [ w / z ] [ y / x ] ps <-> [ y / x ] [ w / z ] ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

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

This theorem is referenced by: (None)