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Theorem ralbi12f 31864
Description: Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
ralbi12f.1  |-  F/_ x A
ralbi12f.2  |-  F/_ x B
Assertion
Ref Expression
ralbi12f  |-  ( ( A  =  B  /\  A. x  e.  A  (
ph 
<->  ps ) )  -> 
( A. x  e.  A  ph  <->  A. x  e.  B  ps )
)

Proof of Theorem ralbi12f
StepHypRef Expression
1 ralbi 2940 . 2  |-  ( A. x  e.  A  ( ph 
<->  ps )  ->  ( A. x  e.  A  ph  <->  A. x  e.  A  ps ) )
2 ralbi12f.1 . . 3  |-  F/_ x A
3 ralbi12f.2 . . 3  |-  F/_ x B
42, 3raleqf 3002 . 2  |-  ( A  =  B  ->  ( A. x  e.  A  ps 
<-> 
A. x  e.  B  ps ) )
51, 4sylan9bbr 701 1  |-  ( ( A  =  B  /\  A. x  e.  A  (
ph 
<->  ps ) )  -> 
( A. x  e.  A  ph  <->  A. x  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   F/_wnfc 2552   A.wral 2756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761
This theorem is referenced by: (None)
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