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Theorem raaan2 27931
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3737. 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 865 . 2  |-  ( ( A  =  (/)  <->  B  =  (/) )  <->  ( ( A  =  (/)  /\  B  =  (/) )  \/  ( -.  A  =  (/)  /\  -.  B  =  (/) ) ) )
2 rzal 3731 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
32adantr 453 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
4 rzal 3731 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
54adantr 453 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  ph )
6 rzal 3731 . . . . 5  |-  ( B  =  (/)  ->  A. y  e.  B  ps )
76adantl 454 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. y  e.  B  ps )
8 pm5.1 832 . . . 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 1183 . . 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 2603 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
11 raaan2.1 . . . . . . 7  |-  F/ y
ph
1211r19.28z 3722 . . . . . 6  |-  ( B  =/=  (/)  ->  ( A. y  e.  B  ( ph  /\  ps )  <->  ( ph  /\ 
A. y  e.  B  ps ) ) )
1312ralbidv 2727 . . . . 5  |-  ( B  =/=  (/)  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  B  ps ) ) )
1410, 13sylbir 206 . . . 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 2603 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
16 nfcv 2574 . . . . . . 7  |-  F/_ x B
17 raaan2.2 . . . . . . 7  |-  F/ x ps
1816, 17nfral 2761 . . . . . 6  |-  F/ x A. y  e.  B  ps
1918r19.27z 3728 . . . . 5  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
2015, 19sylbir 206 . . . 4  |-  ( -.  A  =  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
2114, 20sylan9bbr 683 . . 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 370 . 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 189 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 178    \/ wo 359    /\ wa 360   F/wnf 1554    = wceq 1653    =/= wne 2601   A.wral 2707   (/)c0 3630
This theorem is referenced by:  2reu4a  27945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-nul 3631
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