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Theorem 2ralsng 4020
Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
ralsng.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2ralsng.1  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
2ralsng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A } A. y  e.  { B } ph  <->  ch ) )
Distinct variable groups:    x, A    ps, x    y, A    x, B, y    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)    V( x, y)    W( x, y)

Proof of Theorem 2ralsng
StepHypRef Expression
1 ralsng.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21ralbidv 2839 . . 3  |-  ( x  =  A  ->  ( A. y  e.  { B } ph  <->  A. y  e.  { B } ps ) )
32ralsng 4018 . 2  |-  ( A  e.  V  ->  ( A. x  e.  { A } A. y  e.  { B } ph  <->  A. y  e.  { B } ps ) )
4 2ralsng.1 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54ralsng 4018 . 2  |-  ( B  e.  W  ->  ( A. y  e.  { B } ps  <->  ch ) )
63, 5sylan9bb 711 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A } A. y  e.  { B } ph  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   {csn 3980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-ral 2754  df-v 3059  df-sbc 3280  df-sn 3981
This theorem is referenced by:  mat1ghm  19557  mat1mhm  19558  c0snmgmhm  40187  zrrnghm  40190
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