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Theorem r19.28v 2535
Description: Restricted quantifier version of one direction of 19.28 1525. (The other direction holds when  A is inhabited, see r19.28mv 3423.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
Assertion
Ref Expression
r19.28v  |-  ( (
ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.28v
StepHypRef Expression
1 id 19 . . . 4  |-  ( ph  ->  ph )
21ralrimivw 2481 . . 3  |-  ( ph  ->  A. x  e.  A  ph )
32anim1i 336 . 2  |-  ( (
ph  /\  A. x  e.  A  ps )  ->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
4 r19.26 2533 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4sylibr 133 1  |-  ( (
ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396
This theorem is referenced by:  txlm  12343
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