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Theorem bnj538 29085
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 2561 . . 3  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
2 bnj538.1 . . 3  |-  A  e. 
_V
31, 2bnj524 29082 . 2  |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  [. A  / 
y ]. A. x ( x  e.  B  ->  ph ) )
4 sbcimg 3045 . . . . . 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 29083 . . . . . 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 1556 . . 3  |-  ( A. x [. A  /  y ]. ( x  e.  B  ->  ph )  <->  A. x
( x  e.  B  ->  [. A  /  y ]. ph ) )
10 sbcalg 3052 . . . 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 2561 . . 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 1530    e. wcel 1696   A.wral 2556   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  bnj92  29210  bnj539  29239  bnj540  29240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator