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Theorem sb8h 1808
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8h.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb8h  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb8h
StepHypRef Expression
1 sb8h.1 . 2  |-  ( ph  ->  A. y ph )
21hbsb3 1762 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1727 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvalh 1709 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312   [wsb 1718
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sbhb  1891  sb8euh  1998
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