Theorem sbco4lem 1882

Description: Lemma for sbco4 1883. 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 1863	. . 3 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph )
2	1	sbbii 1648	. 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 1421	. . . . . . 7 |- F/ w ph
4	3	sbco2 1839	. . . . . 6 |- ( [ v / w ] [ w / y ] ph <-> [ v / y ] ph )
5	4	sbbii 1648	. . . . 5 |- ( [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ v / y ] ph )
6	5	sbbii 1648	. . . 4 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ w / v ] [ y / x ] [ v / y ] ph )
7	6	sbbii 1648	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph )
8		nfv 1421	. . . 4 |- F/ w [ y / x ] [ v / y ] ph
9	8	sbco2 1839	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
10	7, 9	bitri 173	. 2 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
11		nfv 1421	. . . . 5 |- F/ v [ w / y ] ph
12	11	sbid2 1730	. . . 4 |- ( [ w / v ] [ v / w ] [ w / y ] ph <-> [ w / y ] ph )
13	12	sbbii 1648	. . 3 |- ( [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / y ] ph )
14	13	sbbii 1648	. 2 |- ( [ x / w ] [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
15	2, 10, 14	3bitr3i 199	1 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] 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-bndl 1399  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:  sbco4  1883