Theorem sbcom2 1906

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 1905	. . . 4 |- ( [ v / z ] [ y / x ] ph <-> [ y / x ] [ v / z ] ph )
2	1	sbbii 1690	. . 3 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / v ] [ y / x ] [ v / z ] ph )
3		sbcom2v2 1905	. . 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 1460	. . 3 |- ( [ y / x ] ph -> A. v [ y / x ] ph )
6	5	sbco2v 1864	. 2 |- ( [ w / v ] [ v / z ] [ y / x ] ph <-> [ w / z ] [ y / x ] ph )
7		ax-17 1460	. . . 4 |- ( ph -> A. v ph )
8	7	sbco2v 1864	. . 3 |- ( [ w / v ] [ v / z ] ph <-> [ w / z ] ph )
9	8	sbbii 1690	. 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 1687

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688

This theorem is referenced by:  2sb5rf  1908  2sb6rf  1909  sbco4lem  1925  sbco4  1926  sbmo  2002  cnvopab  4776