Theorem sbco2h 1838

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
sbco2h.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
sbco2h	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Proof of Theorem sbco2h

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2h.1	. . . . 5 |- ( ph -> A. z ph )
2	1	nfi 1351	. . . 4 |- F/ z ph
3	2	sbco2yz 1837	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
4	3	sbbii 1648	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
5		nfv 1421	. . 3 |- F/ w [ z / x ] ph
6	5	sbco2yz 1837	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
7		nfv 1421	. . 3 |- F/ w ph
8	7	sbco2yz 1837	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
9	4, 6, 8	3bitr3i 199	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbco2  1839  sbco2d  1840  sbco3  1848  elsb3  1852  elsb4  1853  sb9  1855