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Theorem eqrelrd2 25945
Description: A version of eqrelrdv2 4938 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 1811 . . . 4  |-  ( ph  ->  A. y ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) )
51, 4alrimi 1811 . . 3  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
65adantl 466 . 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 25944 . . . . 5  |-  ( Rel 
A  ->  ( A  C_  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
121, 2, 9, 10, 7, 8ssrelf 25944 . . . . 5  |-  ( Rel 
B  ->  ( B  C_  A  <->  A. x A. y
( <. x ,  y
>.  e.  B  ->  <. x ,  y >.  e.  A
) ) )
1311, 12bi2anan9 868 . . . 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 3370 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
15 2albiim 1666 . . . 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 288 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
1716adantr 465 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
186, 17mpbird 232 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2565    C_ wss 3327   <.cop 3882   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-opab 4350  df-xp 4845  df-rel 4846
This theorem is referenced by:  fpwrelmap  26032
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