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Theorem rexcomf 3021
Description: Commutation of restricted 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 1645 . . 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 1798 . . 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 249 . 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 2983 . 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 2983 . 2  |-  ( E. y  e.  B  E. x  e.  A  ph  <->  E. y E. x ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
105, 7, 93bitr4i 277 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 184    /\ wa 369   E.wex 1596    e. wcel 1767   F/_wnfc 2615   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820
This theorem is referenced by:  rexcom  3023  rexcom4f  27052
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