Theorem sbco2yz 1957

Description: This is a version of sbco2 1959 where z is distinct from y. It is a lemma on the way to proving sbco2 1959 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 1940	. . 3 |- F/ z [ y / x ] ph
3	2	nfri 1513	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
4		sbequ 1834	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
5	3, 4	sbieh 1784	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 104   F/wnf 1454   [wsb 1756

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529

This theorem depends on definitions:  df-bi 116  df-nf 1455  df-sb 1757

This theorem is referenced by:  sbco2h  1958