Theorem sbcom2v2 2037

Description: Lemma for proving sbcom2 2038. It is the same as sbcom2v 2036 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 2036	. . 3 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ y / v ] [ w / z ] [ v / x ] ph )
2		sbcom2v 2036	. . . 4 |- ( [ w / z ] [ v / x ] ph <-> [ v / x ] [ w / z ] ph )
3	2	sbbii 1811	. . 3 |- ( [ y / v ] [ w / z ] [ v / x ] ph <-> [ y / v ] [ v / x ] [ w / z ] ph )
4	1, 3	bitri 184	. 2 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ y / v ] [ v / x ] [ w / z ] ph )
5		ax-17 1572	. . . 4 |- ( ph -> A. v ph )
6	5	sbco2vh 1996	. . 3 |- ( [ y / v ] [ v / x ] ph <-> [ y / x ] ph )
7	6	sbbii 1811	. 2 |- ( [ w / z ] [ y / v ] [ v / x ] ph <-> [ w / z ] [ y / x ] ph )
8		ax-17 1572	. . 3 |- ( [ w / z ] ph -> A. v [ w / z ] ph )
9	8	sbco2vh 1996	. 2 |- ( [ y / v ] [ v / x ] [ w / z ] ph <-> [ y / x ] [ w / z ] ph )
10	4, 7, 9	3bitr3i 210	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   <-> wb 105   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbcom2  2038