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Theorem sbalv 1936
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbalv.1  |-  ( [ y  /  x ] ph 
<->  ps )
Assertion
Ref Expression
sbalv  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem sbalv
StepHypRef Expression
1 sbal 1931 . 2  |-  ( [ y  /  x ] A. z ph  <->  A. z [ y  /  x ] ph )
2 sbalv.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32albii 1411 . 2  |-  ( A. z [ y  /  x ] ph  <->  A. z ps )
41, 3bitri 183 1  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1294   [wsb 1699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700
This theorem is referenced by:  sbmo  2014  sbabel  2261  peano2  4438
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