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Theorem raldifeq 3863
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 3626 . . 3  |-  ( A. x  e.  ( A  u.  ( B  \  A
) ) ps  <->  ( A. x  e.  A  ps  /\ 
A. x  e.  ( B  \  A ) ps ) )
42, 3syl6bbr 265 . 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 3854 . . . 4  |-  ( A 
C_  B  <->  ( A  u.  ( B  \  A
) )  =  B )
75, 6sylib 198 . . 3  |-  ( ph  ->  ( A  u.  ( B  \  A ) )  =  B )
87raleqdv 3012 . 2  |-  ( ph  ->  ( A. x  e.  ( A  u.  ( B  \  A ) ) ps  <->  A. x  e.  B  ps ) )
94, 8bitrd 255 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. x  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   A.wral 2756    \ cdif 3413    u. cun 3414    C_ wss 3416
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-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741
This theorem is referenced by:  cantnfrescl  8129  rrxmet  22129
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