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Theorem sbco2h 1915
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 1423 . . . 4  |-  F/ z
ph
32sbco2yz 1914 . . 3  |-  ( [ w  /  z ] [ z  /  x ] ph  <->  [ w  /  x ] ph )
43sbbii 1723 . 2  |-  ( [ y  /  w ] [ w  /  z ] [ z  /  x ] ph  <->  [ y  /  w ] [ w  /  x ] ph )
5 nfv 1493 . . 3  |-  F/ w [ z  /  x ] ph
65sbco2yz 1914 . 2  |-  ( [ y  /  w ] [ w  /  z ] [ z  /  x ] ph  <->  [ y  /  z ] [ z  /  x ] ph )
7 nfv 1493 . . 3  |-  F/ w ph
87sbco2yz 1914 . 2  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
94, 6, 83bitr3i 209 1  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314   [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  sbco2  1916  sbco2d  1917  sbco3  1925  elsb3  1929  elsb4  1930  sb9  1932
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