Theorem sbco3v 1891

Description: Version of sbco3 1896 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbco3v	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sbco3v

Step	Hyp	Ref	Expression
1		nfs1v 1863	. . . 4 |- F/ x [ y / x ] ph
2	1	nfri 1457	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	2	sbco2v 1869	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
4		sbco 1890	. . 3 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
5	4	sbbii 1695	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
6	3, 5	bitr3i 184	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 103   [wsb 1692

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  sbcomv  1893