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Theorem bnj538 28769
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj538.1  |-  A  e. 
_V
Assertion
Ref Expression
bnj538  |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  A. x  e.  B  [. A  / 
y ]. ph )
Distinct variable groups:    x, A    y, B    x, y
Allowed substitution hints:    ph( x, y)    A( y)    B( x)

Proof of Theorem bnj538
StepHypRef Expression
1 df-ral 2548 . . 3  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 bnj538.1 . . 3  |-  A  e. 
_V
31, 2bnj524 28766 . 2  |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  [. A  / 
y ]. A. x ( x  e.  B  ->  ph ) )
4 sbcimg 3032 . . . . . 6  |-  ( A  e.  _V  ->  ( [. A  /  y ]. ( x  e.  B  ->  ph )  <->  ( [. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )
) )
52, 4ax-mp 8 . . . . 5  |-  ( [. A  /  y ]. (
x  e.  B  ->  ph )  <->  ( [. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )
)
62bnj525 28767 . . . . . 6  |-  ( [. A  /  y ]. x  e.  B  <->  x  e.  B
)
76imbi1i 315 . . . . 5  |-  ( (
[. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )  <->  ( x  e.  B  ->  [. A  /  y ]. ph )
)
85, 7bitri 240 . . . 4  |-  ( [. A  /  y ]. (
x  e.  B  ->  ph )  <->  ( x  e.  B  ->  [. A  / 
y ]. ph ) )
98albii 1553 . . 3  |-  ( A. x [. A  /  y ]. ( x  e.  B  ->  ph )  <->  A. x
( x  e.  B  ->  [. A  /  y ]. ph ) )
10 sbcalg 3039 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. A. x ( x  e.  B  ->  ph )  <->  A. x [. A  / 
y ]. ( x  e.  B  ->  ph ) ) )
112, 10ax-mp 8 . . 3  |-  ( [. A  /  y ]. A. x ( x  e.  B  ->  ph )  <->  A. x [. A  /  y ]. ( x  e.  B  ->  ph ) )
12 df-ral 2548 . . 3  |-  ( A. x  e.  B  [. A  /  y ]. ph  <->  A. x
( x  e.  B  ->  [. A  /  y ]. ph ) )
139, 11, 123bitr4i 268 . 2  |-  ( [. A  /  y ]. A. x ( x  e.  B  ->  ph )  <->  A. x  e.  B  [. A  / 
y ]. ph )
143, 13bitri 240 1  |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  A. x  e.  B  [. A  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  bnj92  28894  bnj539  28923  bnj540  28924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992
  Copyright terms: Public domain W3C validator