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Theorem rgen2aOLD 2882
Description: Obsolete proof of rgen2a as of 1-Jan-2020. (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
rgen2a.1  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )
Assertion
Ref Expression
rgen2aOLD  |-  A. x  e.  A  A. y  e.  A  ph
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem rgen2aOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1 2526 . . . . . . . 8  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
2 rgen2a.1 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )
32ex 432 . . . . . . . 8  |-  ( x  e.  A  ->  (
y  e.  A  ->  ph ) )
41, 3syl6bi 228 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  -> 
( y  e.  A  ->  ph ) ) )
54pm2.43d 48 . . . . . 6  |-  ( y  =  x  ->  (
y  e.  A  ->  ph ) )
65alimi 1638 . . . . 5  |-  ( A. y  y  =  x  ->  A. y ( y  e.  A  ->  ph )
)
76a1d 25 . . . 4  |-  ( A. y  y  =  x  ->  ( x  e.  A  ->  A. y ( y  e.  A  ->  ph )
) )
8 eleq1 2526 . . . . . 6  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
98dvelimv 2084 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( x  e.  A  ->  A. y  x  e.  A )
)
103alimi 1638 . . . . 5  |-  ( A. y  x  e.  A  ->  A. y ( y  e.  A  ->  ph )
)
119, 10syl6 33 . . . 4  |-  ( -. 
A. y  y  =  x  ->  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) ) )
127, 11pm2.61i 164 . . 3  |-  ( x  e.  A  ->  A. y
( y  e.  A  ->  ph ) )
13 df-ral 2809 . . 3  |-  ( A. y  e.  A  ph  <->  A. y
( y  e.  A  ->  ph ) )
1412, 13sylibr 212 . 2  |-  ( x  e.  A  ->  A. y  e.  A  ph )
1514rgen 2814 1  |-  A. x  e.  A  A. y  e.  A  ph
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1396    e. wcel 1823   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-cleq 2446  df-clel 2449  df-ral 2809
This theorem is referenced by: (None)
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