Theorem sbco3v 2029

Description: Version of sbco3 2034 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 1999	. . . 4 |- F/ x [ y / x ] ph
2	1	nfri 1572	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	2	sbco2vh 2005	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
4		sbco 2028	. . 3 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
5	4	sbbii 1818	. 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 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbcomv  2031