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Theorem rexcomf 2969
Description: Commutation of restricted existential quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
ralcomf.1  |-  F/_ y A
ralcomf.2  |-  F/_ x B
Assertion
Ref Expression
rexcomf  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem rexcomf
StepHypRef Expression
1 ancom 450 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
21anbi1i 695 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  ph ) )
322exbii 1691 . . 3  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  E. x E. y ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
4 excom 1875 . . 3  |-  ( E. x E. y ( ( y  e.  B  /\  x  e.  A
)  /\  ph )  <->  E. y E. x ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
53, 4bitri 251 . 2  |-  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  E. y E. x ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
6 ralcomf.1 . . 3  |-  F/_ y A
76r2exf 2930 . 2  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
8 ralcomf.2 . . 3  |-  F/_ x B
98r2exf 2930 . 2  |-  ( E. y  e.  B  E. x  e.  A  ph  <->  E. y E. x ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
105, 7, 93bitr4i 279 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369   E.wex 1635    e. wcel 1844   F/_wnfc 2552   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-ex 1636  df-nf 1640  df-sb 1766  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762
This theorem is referenced by:  rexcom  2971  rexcom4f  27802
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