Theorem sbco3v 1957

Description: Version of sbco3 1962 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 1927	. . . 4 |- F/ x [ y / x ] ph
2	1	nfri 1507	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	2	sbco2vh 1933	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
4		sbco 1956	. . 3 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
5	4	sbbii 1753	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
6	3, 5	bitr3i 185	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 104   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbcomv  1959