Theorem sbco3v 1981

Description: Version of sbco3 1986 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 1951	. . . 4 |- F/ x [ y / x ] ph
2	1	nfri 1530	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	2	sbco2vh 1957	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
4		sbco 1980	. . 3 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
5	4	sbbii 1776	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
6	3, 5	bitr3i 186	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 105   [wsb 1773

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbcomv  1983