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Theorem bnj538 29108
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 2710 . . 3  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
21sbcbii 3216 . 2  |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  [. A  / 
y ]. A. x ( x  e.  B  ->  ph ) )
3 bnj538.1 . . . . . 6  |-  A  e. 
_V
4 sbcimg 3202 . . . . . 6  |-  ( A  e.  _V  ->  ( [. A  /  y ]. ( x  e.  B  ->  ph )  <->  ( [. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )
) )
53, 4ax-mp 8 . . . . 5  |-  ( [. A  /  y ]. (
x  e.  B  ->  ph )  <->  ( [. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )
)
63bnj525 29106 . . . . . 6  |-  ( [. A  /  y ]. x  e.  B  <->  x  e.  B
)
76imbi1i 316 . . . . 5  |-  ( (
[. A  /  y ]. x  e.  B  ->  [. A  /  y ]. ph )  <->  ( x  e.  B  ->  [. A  /  y ]. ph )
)
85, 7bitri 241 . . . 4  |-  ( [. A  /  y ]. (
x  e.  B  ->  ph )  <->  ( x  e.  B  ->  [. A  / 
y ]. ph ) )
98albii 1575 . . 3  |-  ( A. x [. A  /  y ]. ( x  e.  B  ->  ph )  <->  A. x
( x  e.  B  ->  [. A  /  y ]. ph ) )
10 sbcal 3208 . . 3  |-  ( [. A  /  y ]. A. x ( x  e.  B  ->  ph )  <->  A. x [. A  /  y ]. ( x  e.  B  ->  ph ) )
11 df-ral 2710 . . 3  |-  ( A. x  e.  B  [. A  /  y ]. ph  <->  A. x
( x  e.  B  ->  [. A  /  y ]. ph ) )
129, 10, 113bitr4i 269 . 2  |-  ( [. A  /  y ]. A. x ( x  e.  B  ->  ph )  <->  A. x  e.  B  [. A  / 
y ]. ph )
132, 12bitri 241 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 177   A.wal 1549    e. wcel 1725   A.wral 2705   _Vcvv 2956   [.wsbc 3161
This theorem is referenced by:  bnj92  29233  bnj539  29262  bnj540  29263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-sbc 3162
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