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Theorem bj-rexcom4a 31547
Description: Remove from rexcom4a 3054 dependency on ax-ext 2451 and ax-13 2104 (and on df-or 377, df-sb 1806, df-clab 2458, df-cleq 2464, df-clel 2467, df-nfc 2601, df-v 3033). This proof uses only df-rex 2762 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-rexcom4a  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Distinct variable groups:    x, A    x, y    ph, x
Allowed substitution hints:    ph( y)    ps( x, y)    A( y)

Proof of Theorem bj-rexcom4a
StepHypRef Expression
1 bj-rexcom4 31546 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. x E. y  e.  A  ( ph  /\ 
ps ) )
2 19.42v 1842 . . 3  |-  ( E. x ( ph  /\  ps )  <->  ( ph  /\  E. x ps ) )
32rexbii 2881 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
41, 3bitr3i 259 1  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   E.wex 1671   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-11 1937
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-rex 2762
This theorem is referenced by:  bj-rexcom4bv  31548  bj-rexcom4b  31549
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