Theorem sbco2h 2017

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 1510	. . . 4 |- F/ z ph
3	2	sbco2yz 2016	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
4	3	sbbii 1813	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
5		nfv 1576	. . 3 |- F/ w [ z / x ] ph
6	5	sbco2yz 2016	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
7		nfv 1576	. . 3 |- F/ w ph
8	7	sbco2yz 2016	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
9	4, 6, 8	3bitr3i 210	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   [wsb 1810

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sbco2  2018  sbco2d  2019  sbco3  2027  sb9  2032  elsb1  2209  elsb2  2210