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Theorem r19.36zv 3496
Description: Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
r19.36zv  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  ps ) ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem r19.36zv
StepHypRef Expression
1 r19.9rzv 3490 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  E. x  e.  A  ps ) )
21imbi2d 309 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) ) )
3 r19.35 2658 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
42, 3syl6rbbr 257 1  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    =/= wne 2419   A.wral 2516   E.wrex 2517   (/)c0 3397
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-v 2742  df-dif 3097  df-nul 3398
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