Theorem sbcom2 2445

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.)

Assertion

Ref	Expression
sbcom2	|- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Distinct variable groups:   x, z   x, w   y, z

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem sbcom2

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		2sb6 2444	. . . . . . . . . 10 |- ( [ v / z ] [ u / x ] ph <-> A. z A. x ( ( z = v /\ x = u ) -> ph ) )
4		alcom 2037	. . . . . . . . . 10 |- ( A. z A. x ( ( z = v /\ x = u ) -> ph ) <-> A. x A. z ( ( z = v /\ x = u ) -> ph ) )
5		ancomst 468	. . . . . . . . . . 11 |- ( ( ( z = v /\ x = u ) -> ph ) <-> ( ( x = u /\ z = v ) -> ph ) )
6	5	2albii 1748	. . . . . . . . . 10 |- ( A. x A. z ( ( z = v /\ x = u ) -> ph ) <-> A. x A. z ( ( x = u /\ z = v ) -> ph ) )
7	3, 4, 6	3bitri 286	. . . . . . . . 9 |- ( [ v / z ] [ u / x ] ph <-> A. x A. z ( ( x = u /\ z = v ) -> ph ) )
8		2sb6 2444	. . . . . . . . 9 |- ( [ u / x ] [ v / z ] ph <-> A. x A. z ( ( x = u /\ z = v ) -> ph ) )
9	7, 8	bitr4i 267	. . . . . . . 8 |- ( [ v / z ] [ u / x ] ph <-> [ u / x ] [ v / z ] ph )
10		nfv 1843	. . . . . . . . 9 |- F/ z u = y
11		sbequ 2376	. . . . . . . . 9 |- ( u = y -> ( [ u / x ] ph <-> [ y / x ] ph ) )
12	10, 11	sbbid 2403	. . . . . . . 8 |- ( u = y -> ( [ v / z ] [ u / x ] ph <-> [ v / z ] [ y / x ] ph ) )
13	9, 12	syl5bbr 274	. . . . . . 7 |- ( u = y -> ( [ u / x ] [ v / z ] ph <-> [ v / z ] [ y / x ] ph ) )
14		sbequ 2376	. . . . . . 7 |- ( v = w -> ( [ v / z ] [ y / x ] ph <-> [ w / z ] [ y / x ] ph ) )
15	13, 14	sylan9bb 736	. . . . . 6 |- ( ( u = y /\ v = w ) -> ( [ u / x ] [ v / z ] ph <-> [ w / z ] [ y / x ] ph ) )
16		nfv 1843	. . . . . . . 8 |- F/ x v = w
17		sbequ 2376	. . . . . . . 8 |- ( v = w -> ( [ v / z ] ph <-> [ w / z ] ph ) )
18	16, 17	sbbid 2403	. . . . . . 7 |- ( v = w -> ( [ u / x ] [ v / z ] ph <-> [ u / x ] [ w / z ] ph ) )
19		sbequ 2376	. . . . . . 7 |- ( u = y -> ( [ u / x ] [ w / z ] ph <-> [ y / x ] [ w / z ] ph ) )
20	18, 19	sylan9bbr 737	. . . . . 6 |- ( ( u = y /\ v = w ) -> ( [ u / x ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph ) )
21	15, 20	bitr3d 270	. . . . 5 |- ( ( u = y /\ v = w ) -> ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) )
22	21	ex 450	. . . 4 |- ( u = y -> ( v = w -> ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) ) )
23	22	exlimdv 1861	. . 3 |- ( u = y -> ( E. v v = w -> ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) ) )
24	23	exlimiv 1858	. 2 |- ( E. u u = y -> ( E. v v = w -> ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) ) )
25	1, 2, 24	mp2 9	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   -> 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-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco4lem  2465  sbco4  2466  2mo  2551  cnvopab  5533