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Theorem reusv3i 4600
Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
reusv3.2  |-  ( y  =  z  ->  C  =  D )
Assertion
Ref Expression
reusv3i  |-  ( E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
Distinct variable groups:    x, y,
z, B    x, C, z    x, D, y    ph, x, z    ps, x, y
Allowed substitution hints:    ph( y)    ps( z)    A( x, y, z)    C( y)    D( z)

Proof of Theorem reusv3i
StepHypRef Expression
1 reusv3.1 . . . . . 6  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
2 reusv3.2 . . . . . . 7  |-  ( y  =  z  ->  C  =  D )
32eqeq2d 2416 . . . . . 6  |-  ( y  =  z  ->  (
x  =  C  <->  x  =  D ) )
41, 3imbi12d 318 . . . . 5  |-  ( y  =  z  ->  (
( ph  ->  x  =  C )  <->  ( ps  ->  x  =  D ) ) )
54cbvralv 3033 . . . 4  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  <->  A. z  e.  B  ( ps  ->  x  =  D ) )
65biimpi 194 . . 3  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  A. z  e.  B  ( ps  ->  x  =  D ) )
7 raaanv 3881 . . . 4  |-  ( A. y  e.  B  A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  <->  ( A. y  e.  B  ( ph  ->  x  =  C )  /\  A. z  e.  B  ( ps  ->  x  =  D ) ) )
8 prth 569 . . . . . . 7  |-  ( ( ( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  (
( ph  /\  ps )  ->  ( x  =  C  /\  x  =  D ) ) )
9 eqtr2 2429 . . . . . . 7  |-  ( ( x  =  C  /\  x  =  D )  ->  C  =  D )
108, 9syl6 31 . . . . . 6  |-  ( ( ( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  (
( ph  /\  ps )  ->  C  =  D ) )
1110ralimi 2796 . . . . 5  |-  ( A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
1211ralimi 2796 . . . 4  |-  ( A. y  e.  B  A. z  e.  B  (
( ph  ->  x  =  C )  /\  ( ps  ->  x  =  D ) )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
137, 12sylbir 213 . . 3  |-  ( ( A. y  e.  B  ( ph  ->  x  =  C )  /\  A. z  e.  B  ( ps  ->  x  =  D ) )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
146, 13mpdan 666 . 2  |-  ( A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
1514rexlimivw 2892 1  |-  ( E. x  e.  A  A. y  e.  B  ( ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   A.wral 2753   E.wrex 2754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-nul 3738
This theorem is referenced by:  reusv3  4601
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