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Theorem raldifeq 3768
Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
Hypotheses
Ref Expression
raldifeq.1  |-  ( ph  ->  A  C_  B )
raldifeq.2  |-  ( ph  ->  A. x  e.  ( B  \  A ) ps )
Assertion
Ref Expression
raldifeq  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem raldifeq
StepHypRef Expression
1 raldifeq.2 . . . 4  |-  ( ph  ->  A. x  e.  ( B  \  A ) ps )
21biantrud 507 . . 3  |-  ( ph  ->  ( A. x  e.  A  ps  <->  ( A. x  e.  A  ps  /\ 
A. x  e.  ( B  \  A ) ps ) ) )
3 ralunb 3537 . . 3  |-  ( A. x  e.  ( A  u.  ( B  \  A
) ) ps  <->  ( A. x  e.  A  ps  /\ 
A. x  e.  ( B  \  A ) ps ) )
42, 3syl6bbr 263 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  ( A  u.  ( B  \  A ) ) ps ) )
5 raldifeq.1 . . . 4  |-  ( ph  ->  A  C_  B )
6 undif 3759 . . . 4  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  =  B )
75, 6sylib 196 . . 3  |-  ( ph  ->  ( A  u.  ( B  \  A ) )  =  B )
87raleqdv 2923 . 2  |-  ( ph  ->  ( A. x  e.  ( A  u.  ( B  \  A ) ) ps  <->  A. x  e.  B  ps ) )
94, 8bitrd 253 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   A.wral 2715    \ cdif 3325    u. cun 3326    C_ wss 3328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638
This theorem is referenced by:  rrxmet  20907
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