Theorem sbcom2v2 1862

Description: Lemma for proving sbcom2 1863. It is the same as sbcom2v 1861 but removes the distinct variable constraint on x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2v2	|- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

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

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

Proof of Theorem sbcom2v2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v 1861	. . 3 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ y / v ] [ w / z ] [ v / x ] ph )
2		sbcom2v 1861	. . . 4 |- ( [ w / z ] [ v / x ] ph <-> [ v / x ] [ w / z ] ph )
3	2	sbbii 1648	. . 3 |- ( [ y / v ] [ w / z ] [ v / x ] ph <-> [ y / v ] [ v / x ] [ w / z ] ph )
4	1, 3	bitri 173	. 2 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ y / v ] [ v / x ] [ w / z ] ph )
5		ax-17 1419	. . . 4 |- ( ph -> A. v ph )
6	5	sbco2v 1821	. . 3 |- ( [ y / v ] [ v / x ] ph <-> [ y / x ] ph )
7	6	sbbii 1648	. 2 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ w / z ] [ y / x ] ph )
8		ax-17 1419	. . 3 |- ( [ w / z ] ph -> A. v [ w / z ] ph )
9	8	sbco2v 1821	. 2 |- ( [ y / v ] [ v / x ] [ w / z ] ph <-> [ y / x ] [ w / z ] ph )
10	4, 7, 9	3bitr3i 199	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   <-> wb 98   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbcom2  1863