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Theorem bj-rexcom4b 34591
Description: Remove from rexcom4b 3131 dependency on ax-ext 2435 and ax-13 2000 (and on df-or 370, df-cleq 2449, df-nfc 2607, df-v 3111). The hypothesis uses  V instead of  _V (see bj-isseti 34583 for the motivation). Use bj-rexcom4bv 34590 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4b.1  |-  B  e.  V
Assertion
Ref Expression
bj-rexcom4b  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Distinct variable groups:    x, A    x, B    x, y    ph, x
Allowed substitution hints:    ph( y)    A( y)    B( y)    V( x, y)

Proof of Theorem bj-rexcom4b
StepHypRef Expression
1 bj-rexcom4a 34589 . 2  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
2 bj-rexcom4b.1 . . . . 5  |-  B  e.  V
32bj-isseti 34583 . . . 4  |-  E. x  x  =  B
43biantru 505 . . 3  |-  ( ph  <->  (
ph  /\  E. x  x  =  B )
)
54rexbii 2959 . 2  |-  ( E. y  e.  A  ph  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
61, 5bitr4i 252 1  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   E.wrex 2808
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-11 1843  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-sb 1741  df-clab 2443  df-clel 2452  df-rex 2813
This theorem is referenced by: (None)
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