Theorem sbco2h 2024

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 1515	. . . 4 |- F/ z ph
3	2	sbco2yz 2023	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
4	3	sbbii 1818	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
5		nfv 1581	. . 3 |- F/ w [ z / x ] ph
6	5	sbco2yz 2023	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
7		nfv 1581	. . 3 |- F/ w ph
8	7	sbco2yz 2023	. 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 1400   [wsb 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbco2  2025  sbco2d  2026  sbco3  2034  sb9  2039  elsb1  2216  elsb2  2217