Theorem sbco2yz 1910

Description: This is a version of sbco2 1912 where z is distinct from y. It is a lemma on the way to proving sbco2 1912 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 1895	. . 3 |- F/ z [ y / x ] ph
3	2	nfri 1480	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
4		sbequ 1792	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
5	3, 4	sbieh 1744	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 104   F/wnf 1417   [wsb 1716

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717

This theorem is referenced by:  sbco2h  1911