Theorem sbco2 1984

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sbco2.1	|- F/ z ph

Assertion

Ref	Expression
sbco2	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Proof of Theorem sbco2

Step	Hyp	Ref	Expression
1		sbco2.1	. . 3 |- F/ z ph
2	1	nfri 1533	. 2 |- ( ph -> A. z ph )
3	2	sbco2h 1983	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 105   F/wnf 1474   [wsb 1776

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  nfsbt  1995  sb7af  2012  sbco4lem  2025  sbco4  2026  eqsb1  2300  clelsb1  2301  clelsb2  2302  sb8ab  2318  clelsb1f  2343  sbralie  2747  sbcco  3011