Theorem sbco4lem 2465

Description: Lemma for sbco4 2466. 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 2445	. . 3 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph )
2	1	sbbii 1887	. 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 1843	. . . . . . 7 |- F/ w ph
4	3	sbco2 2415	. . . . . 6 |- ( [ v / w ] [ w / y ] ph <-> [ v / y ] ph )
5	4	sbbii 1887	. . . . 5 |- ( [ y / x ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ v / y ] ph )
6	5	sbbii 1887	. . . 4 |- ( [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ w / v ] [ y / x ] [ v / y ] ph )
7	6	sbbii 1887	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph )
8		nfv 1843	. . . 4 |- F/ w [ y / x ] [ v / y ] ph
9	8	sbco2 2415	. . 3 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
10	7, 9	bitri 264	. 2 |- ( [ x / w ] [ w / v ] [ y / x ] [ v / w ] [ w / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
11		sbid2v 2457	. . . 4 |- ( [ w / v ] [ v / w ] [ w / y ] ph <-> [ w / y ] ph )
12	11	sbbii 1887	. . 3 |- ( [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ y / x ] [ w / y ] ph )
13	12	sbbii 1887	. 2 |- ( [ x / w ] [ y / x ] [ w / v ] [ v / w ] [ w / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
14	2, 10, 13	3bitr3i 290	1 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Syntax hints:   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco4  2466