Theorem sbco2h 1964

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 1462	. . . 4 |- F/ z ph
3	2	sbco2yz 1963	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
4	3	sbbii 1765	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
5		nfv 1528	. . 3 |- F/ w [ z / x ] ph
6	5	sbco2yz 1963	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
7		nfv 1528	. . 3 |- F/ w ph
8	7	sbco2yz 1963	. 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 1351   [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbco2  1965  sbco2d  1966  sbco3  1974  sb9  1979  elsb1  2155  elsb2  2156