Theorem sbco2yz 1991

Description: This is a version of sbco2 1993 where z is distinct from y. It is a lemma on the way to proving sbco2 1993 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
sbco2yz.1	|- F/ z ph

Assertion

Ref	Expression
sbco2yz	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Distinct variable group:   y, z

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

Proof of Theorem sbco2yz

Step	Hyp	Ref	Expression
1		sbco2yz.1	. . . 4 |- F/ z ph
2	1	nfsb 1974	. . 3 |- F/ z [ y / x ] ph
3	2	nfri 1542	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
4		sbequ 1863	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
5	3, 4	sbieh 1813	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 105   F/wnf 1483   [wsb 1785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbco2h  1992