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Theorem cbvex2OLD 2030
Description: Obsolete proof of cbvex2 2029 as of 16-Jun-2019. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cbval2.1  |-  F/ z
ph
cbval2.2  |-  F/ w ph
cbval2.3  |-  F/ x ps
cbval2.4  |-  F/ y ps
cbval2.5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvex2OLD  |-  ( E. x E. y ph  <->  E. z E. w ps )
Distinct variable groups:    x, y    y, z    x, w    z, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)

Proof of Theorem cbvex2OLD
StepHypRef Expression
1 cbval2.1 . . 3  |-  F/ z
ph
21nfex 1949 . 2  |-  F/ z E. y ph
3 cbval2.3 . . 3  |-  F/ x ps
43nfex 1949 . 2  |-  F/ x E. w ps
5 nfv 1708 . . . . . 6  |-  F/ w  x  =  z
6 cbval2.2 . . . . . 6  |-  F/ w ph
75, 6nfan 1929 . . . . 5  |-  F/ w
( x  =  z  /\  ph )
8 nfv 1708 . . . . . 6  |-  F/ y  x  =  z
9 cbval2.4 . . . . . 6  |-  F/ y ps
108, 9nfan 1929 . . . . 5  |-  F/ y ( x  =  z  /\  ps )
11 cbval2.5 . . . . . . 7  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
1211expcom 435 . . . . . 6  |-  ( y  =  w  ->  (
x  =  z  -> 
( ph  <->  ps ) ) )
1312pm5.32d 639 . . . . 5  |-  ( y  =  w  ->  (
( x  =  z  /\  ph )  <->  ( x  =  z  /\  ps )
) )
147, 10, 13cbvex 2023 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  E. w ( x  =  z  /\  ps )
)
15 19.42v 1776 . . . 4  |-  ( E. y ( x  =  z  /\  ph )  <->  ( x  =  z  /\  E. y ph ) )
16 19.42v 1776 . . . 4  |-  ( E. w ( x  =  z  /\  ps )  <->  ( x  =  z  /\  E. w ps ) )
1714, 15, 163bitr3i 275 . . 3  |-  ( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) )
18 pm5.32 636 . . 3  |-  ( ( x  =  z  -> 
( E. y ph  <->  E. w ps ) )  <-> 
( ( x  =  z  /\  E. y ph )  <->  ( x  =  z  /\  E. w ps ) ) )
1917, 18mpbir 209 . 2  |-  ( x  =  z  ->  ( E. y ph  <->  E. w ps ) )
202, 4, 19cbvex 2023 1  |-  ( E. x E. y ph  <->  E. z E. w ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1613   F/wnf 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618
This theorem is referenced by: (None)
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