Theorem sbcom2 1904

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 modified to be intuitionistic by Jim Kingdon, 19-Feb-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 variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v2 1903	. . . 4 |- ( [ v / z ] [ y / x ] ph <-> [ y / x ] [ v / z ] ph )
2	1	sbbii 1688	. . 3 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / v ] [ y / x ] [ v / z ] ph )
3		sbcom2v2 1903	. . 3 |- ( [ w / v ] [ y / x ] [ v / z ] ph <-> [ y / x ] [ w / v ] [ v / z ] ph )
4	2, 3	bitri 182	. 2 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ y / x ] [ w / v ] [ v / z ] ph )
5		ax-17 1459	. . 3 |- ( [ y / x ] ph -> A. v [ y / x ] ph )
6	5	sbco2v 1862	. 2 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / z ] [ y / x ] ph )
7		ax-17 1459	. . . 4 |- ( ph -> A. v ph )
8	7	sbco2v 1862	. . 3 |- ( [ w / v ] [ v / z ] ph <-> [ w / z ] ph )
9	8	sbbii 1688	. 2 |- ( [ y / x ] [ w / v ] [ v / z ] ph <-> [ y / x ] [ w / z ] ph )
10	4, 6, 9	3bitr3i 208	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  2sb5rf  1906  2sb6rf  1907  sbco4lem  1923  sbco4  1924  sbmo  2000  cnvopab  4746