Theorem sbco3 2002

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Assertion

Ref	Expression
sbco3	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Proof of Theorem sbco3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco3xzyz 2001	. . 3 |- ( [ w / y ] [ y / x ] ph <-> [ w / x ] [ x / y ] ph )
2	1	sbbii 1788	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / w ] [ w / x ] [ x / y ] ph )
3		ax-17 1549	. . 3 |- ( [ y / x ] ph -> A. w [ y / x ] ph )
4	3	sbco2h 1992	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
5		ax-17 1549	. . 3 |- ( [ x / y ] ph -> A. w [ x / y ] ph )
6	5	sbco2h 1992	. 2 |- ( [ z / w ] [ w / x ] [ x / y ] ph <-> [ z / x ] [ x / y ] ph )
7	2, 4, 6	3bitr3i 210	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 105   [wsb 1785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbcom  2003