Theorem sbco2h 1937

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 1438	. . . 4 |- F/ z ph
3	2	sbco2yz 1936	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
4	3	sbbii 1738	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
5		nfv 1508	. . 3 |- F/ w [ z / x ] ph
6	5	sbco2yz 1936	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
7		nfv 1508	. . 3 |- F/ w ph
8	7	sbco2yz 1936	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
9	4, 6, 8	3bitr3i 209	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbco2  1938  sbco2d  1939  sbco3  1947  elsb3  1951  elsb4  1952  sb9  1954