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Theorem bj-rexcom4 31546
Description: Remove from rexcom4 3053 dependency on ax-ext 2451 and ax-13 2104 (and on df-or 377, df-tru 1455, 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, 13-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-rexcom4  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem bj-rexcom4
StepHypRef Expression
1 df-rex 2762 . 2  |-  ( E. x  e.  A  E. y ph  <->  E. x ( x  e.  A  /\  E. y ph ) )
2 19.42v 1842 . . . . 5  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
32bicomi 207 . . . 4  |-  ( ( x  e.  A  /\  E. y ph )  <->  E. y
( x  e.  A  /\  ph ) )
43exbii 1726 . . 3  |-  ( E. x ( x  e.  A  /\  E. y ph )  <->  E. x E. y
( x  e.  A  /\  ph ) )
5 excom 1944 . . . 4  |-  ( E. x E. y ( x  e.  A  /\  ph )  <->  E. y E. x
( x  e.  A  /\  ph ) )
6 df-rex 2762 . . . . . 6  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
76bicomi 207 . . . . 5  |-  ( E. x ( x  e.  A  /\  ph )  <->  E. x  e.  A  ph )
87exbii 1726 . . . 4  |-  ( E. y E. x ( x  e.  A  /\  ph )  <->  E. y E. x  e.  A  ph )
95, 8bitri 257 . . 3  |-  ( E. x E. y ( x  e.  A  /\  ph )  <->  E. y E. x  e.  A  ph )
104, 9bitri 257 . 2  |-  ( E. x ( x  e.  A  /\  E. y ph )  <->  E. y E. x  e.  A  ph )
111, 10bitri 257 1  |-  ( E. x  e.  A  E. y ph  <->  E. y E. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   E.wex 1671    e. wcel 1904   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-rexcom4a  31547
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