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Theorem eqrelrd2 28061
Description: A version of eqrelrdv2 4954 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
eqrelrd2.1  |-  F/ x ph
eqrelrd2.2  |-  F/ y
ph
eqrelrd2.3  |-  F/_ x A
eqrelrd2.4  |-  F/_ y A
eqrelrd2.5  |-  F/_ x B
eqrelrd2.6  |-  F/_ y B
eqrelrd2.7  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrd2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem eqrelrd2
StepHypRef Expression
1 eqrelrd2.1 . . . 4  |-  F/ x ph
2 eqrelrd2.2 . . . . 5  |-  F/ y
ph
3 eqrelrd2.7 . . . . 5  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
42, 3alrimi 1930 . . . 4  |-  ( ph  ->  A. y ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) )
51, 4alrimi 1930 . . 3  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
65adantl 467 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
7 eqrelrd2.3 . . . . . 6  |-  F/_ x A
8 eqrelrd2.4 . . . . . 6  |-  F/_ y A
9 eqrelrd2.5 . . . . . 6  |-  F/_ x B
10 eqrelrd2.6 . . . . . 6  |-  F/_ y B
111, 2, 7, 8, 9, 10ssrelf 28060 . . . . 5  |-  ( Rel 
A  ->  ( A  C_  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
121, 2, 9, 10, 7, 8ssrelf 28060 . . . . 5  |-  ( Rel 
B  ->  ( B  C_  A  <->  A. x A. y
( <. x ,  y
>.  e.  B  ->  <. x ,  y >.  e.  A
) ) )
1311, 12bi2anan9 881 . . . 4  |-  ( ( Rel  A  /\  Rel  B )  ->  ( ( A  C_  B  /\  B  C_  A )  <->  ( A. x A. y ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)  /\  A. x A. y ( <. x ,  y >.  e.  B  -> 
<. x ,  y >.  e.  A ) ) ) )
14 eqss 3485 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
15 2albiim 1747 . . . 4  |-  ( A. x A. y ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )  <->  ( A. x A. y ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)  /\  A. x A. y ( <. x ,  y >.  e.  B  -> 
<. x ,  y >.  e.  A ) ) )
1613, 14, 153bitr4g 291 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
1716adantr 466 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
186, 17mpbird 235 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1663    e. wcel 1870   F/_wnfc 2577    C_ wss 3442   <.cop 4008   Rel wrel 4859
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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  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-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-opab 4485  df-xp 4860  df-rel 4861
This theorem is referenced by:  fpwrelmap  28161
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