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Theorem raaan2 37987
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3911. 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 901 . 2  |-  ( ( A  =  (/)  <->  B  =  (/) )  <->  ( ( A  =  (/)  /\  B  =  (/) )  \/  ( -.  A  =  (/)  /\  -.  B  =  (/) ) ) )
2 rzal 3905 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
32adantr 466 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  A. y  e.  B  ( ph  /\ 
ps ) )
4 rzal 3905 . . . . 5  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
54adantr 466 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. x  e.  A  ph )
6 rzal 3905 . . . . 5  |-  ( B  =  (/)  ->  A. y  e.  B  ps )
76adantl 467 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A. y  e.  B  ps )
8 pm5.1 865 . . . 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 1262 . . 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 2627 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
11 raaan2.1 . . . . . . 7  |-  F/ y
ph
1211r19.28z 3895 . . . . . 6  |-  ( B  =/=  (/)  ->  ( A. y  e.  B  ( ph  /\  ps )  <->  ( ph  /\ 
A. y  e.  B  ps ) ) )
1312ralbidv 2871 . . . . 5  |-  ( B  =/=  (/)  ->  ( A. x  e.  A  A. y  e.  B  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  B  ps ) ) )
1410, 13sylbir 216 . . . 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 2627 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
16 nfcv 2591 . . . . . . 7  |-  F/_ x B
17 raaan2.2 . . . . . . 7  |-  F/ x ps
1816, 17nfral 2818 . . . . . 6  |-  F/ x A. y  e.  B  ps
1918r19.27z 3902 . . . . 5  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph  /\  A. y  e.  B  ps ) ) )
2015, 19sylbir 216 . . . 4  |-  ( -.  A  =  (/)  ->  ( A. x  e.  A  ( ph  /\  A. y  e.  B  ps )  <->  ( A. x  e.  A  ph 
/\  A. y  e.  B  ps ) ) )
2114, 20sylan9bbr 705 . . 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 380 . 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 198 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   F/wnf 1663    =/= wne 2625   A.wral 2782   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-v 3089  df-dif 3445  df-nul 3768
This theorem is referenced by:  2reu4a  38001
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