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Theorem sbco2h 1881
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.
StepHypRef Expression
1 sbco2h.1 . . . . 5  |-  ( ph  ->  A. z ph )
21nfi 1392 . . . 4  |-  F/ z
ph
32sbco2yz 1880 . . 3  |-  ( [ w  /  z ] [ z  /  x ] ph  <->  [ w  /  x ] ph )
43sbbii 1690 . 2  |-  ( [ y  /  w ] [ w  /  z ] [ z  /  x ] ph  <->  [ y  /  w ] [ w  /  x ] ph )
5 nfv 1462 . . 3  |-  F/ w [ z  /  x ] ph
65sbco2yz 1880 . 2  |-  ( [ y  /  w ] [ w  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph )
7 nfv 1462 . . 3  |-  F/ w ph
87sbco2yz 1880 . 2  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
94, 6, 83bitr3i 208 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbco2  1882  sbco2d  1883  sbco3  1891  elsb3  1895  elsb4  1896  sb9  1898
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