Theorem sbco4lem 1923

Description: Lemma for sbco4 1924. It replaces the temporary variable v with another temporary variable w. (Contributed by Jim Kingdon, 26-Sep-2018.)

Assertion

Ref	Expression
sbco4lem	|- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Distinct variable groups:   w, v, ph   x, v, w   y, v, w

Allowed substitution hints:   ph( x, y)

Proof of Theorem sbco4lem

Step	Hyp	Ref	Expression
1		sbcom2 1904	. . 3 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph )
2	1	sbbii 1688	. 2 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph )
3		nfv 1461	. . . . . . 7 |- F/ w ph
4	3	sbco2 1880	. . . . . 6 |- ( [ v / w ] [ w / y ] ph <-> [ v / y ] ph )
5	4	sbbii 1688	. . . . 5 |- ( [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ v / y ] ph )
6	5	sbbii 1688	. . . 4 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ w / v ] [ y / x ] [ v / y ] ph )
7	6	sbbii 1688	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph )
8		nfv 1461	. . . 4 |- F/ w [ y / x ] [ v / y ] ph
9	8	sbco2 1880	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
10	7, 9	bitri 182	. 2 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
11		nfv 1461	. . . . 5 |- F/ v [ w / y ] ph
12	11	sbid2 1771	. . . 4 |- ( [ w / v ] [ v / w ] [ w / y ] ph <-> [ w / y ] ph )
13	12	sbbii 1688	. . 3 |- ( [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / y ] ph )
14	13	sbbii 1688	. 2 |- ( [ x / w ] [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
15	2, 10, 14	3bitr3i 208	1 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] 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-bndl 1439  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:  sbco4  1924