Theorem sbco4lem 2016

Description: Lemma for sbco4 2017. 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 1997	. . 3 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph )
2	1	sbbii 1775	. 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 1538	. . . . . . 7 |- F/ w ph
4	3	sbco2 1975	. . . . . 6 |- ( [ v / w ] [ w / y ] ph <-> [ v / y ] ph )
5	4	sbbii 1775	. . . . 5 |- ( [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ v / y ] ph )
6	5	sbbii 1775	. . . 4 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ w / v ] [ y / x ] [ v / y ] ph )
7	6	sbbii 1775	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph )
8		nfv 1538	. . . 4 |- F/ w [ y / x ] [ v / y ] ph
9	8	sbco2 1975	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
10	7, 9	bitri 184	. 2 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
11		nfv 1538	. . . . 5 |- F/ v [ w / y ] ph
12	11	sbid2 1860	. . . 4 |- ( [ w / v ] [ v / w ] [ w / y ] ph <-> [ w / y ] ph )
13	12	sbbii 1775	. . 3 |- ( [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / y ] ph )
14	13	sbbii 1775	. 2 |- ( [ x / w ] [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
15	2, 10, 14	3bitr3i 210	1 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Syntax hints:   <-> wb 105   [wsb 1772

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773

This theorem is referenced by:  sbco4  2017