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Theorem raaan2 27820
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3695. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Hypotheses
Ref Expression
raaan2.1  |-  F/ y
ph
raaan2.2  |-  F/ x ps
Assertion
Ref Expression
raaan2  |-  ( ( A  =  (/)  <->  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem raaan2
StepHypRef Expression
1 dfbi3 864 . 2  |-  ( ( A  =  (/)  <->  B  =  (/) )  <->  ( ( A  =  (/)  /\  B  =  (/) )  \/  ( -.  A  =  (/)  /\  -.  B  =  (/) ) ) )
2 rzal 3689 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
32adantr 452 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
4 rzal 3689 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
54adantr 452 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  ph )
6 rzal 3689 . . . . 5  |-  ( B  =  (/)  ->  A. y  e.  B  ps )
76adantl 453 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. y  e.  B  ps )
8 pm5.1 831 . . . 4  |-  ( ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  /\  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
93, 5, 7, 8syl12anc 1182 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
10 df-ne 2569 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
11 raaan2.1 . . . . . . 7  |-  F/ y
ph
1211r19.28z 3680 . . . . . 6  |-  ( B  =/=  (/)  ->  ( A. y  e.  B  ( ph  /\  ps )  <->  ( ph  /\ 
A. y  e.  B  ps ) ) )
1312ralbidv 2686 . . . . 5  |-  ( B  =/=  (/)  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  B  ps ) ) )
1410, 13sylbir 205 . . . 4  |-  ( -.  B  =  (/)  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  A. x  e.  A  (
ph  /\  A. y  e.  B  ps )
) )
15 df-ne 2569 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
16 nfcv 2540 . . . . . . 7  |-  F/_ x B
17 raaan2.2 . . . . . . 7  |-  F/ x ps
1816, 17nfral 2719 . . . . . 6  |-  F/ x A. y  e.  B  ps
1918r19.27z 3686 . . . . 5  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
2015, 19sylbir 205 . . . 4  |-  ( -.  A  =  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
2114, 20sylan9bbr 682 . . 3  |-  ( ( -.  A  =  (/)  /\ 
-.  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
229, 21jaoi 369 . 2  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( -.  A  =  (/)  /\  -.  B  =  (/) ) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
231, 22sylbi 188 1  |-  ( ( A  =  (/)  <->  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   F/wnf 1550    = wceq 1649    =/= wne 2567   A.wral 2666   (/)c0 3588
This theorem is referenced by:  2reu4a  27834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-v 2918  df-dif 3283  df-nul 3589
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