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Theorem raaan2 27828
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3703. 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 3697 . . . . 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 3697 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
54adantr 452 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  ph )
6 rzal 3697 . . . . 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 2577 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
11 raaan2.1 . . . . . . 7  |-  F/ y
ph
1211r19.28z 3688 . . . . . 6  |-  ( B  =/=  (/)  ->  ( A. y  e.  B  ( ph  /\  ps )  <->  ( ph  /\ 
A. y  e.  B  ps ) ) )
1312ralbidv 2694 . . . . 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 2577 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
16 nfcv 2548 . . . . . . 7  |-  F/_ x B
17 raaan2.2 . . . . . . 7  |-  F/ x ps
1816, 17nfral 2727 . . . . . 6  |-  F/ x A. y  e.  B  ps
1918r19.27z 3694 . . . . 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 2575   A.wral 2674   (/)c0 3596
This theorem is referenced by:  2reu4a  27842
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 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-v 2926  df-dif 3291  df-nul 3597
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