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Theorem rexcomf 2699
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 437 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  <->  ( y  e.  B  /\  x  e.  A )
)
21anbi1i 676 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  /\  ph ) )
322exbii 1570 . . 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 1786 . . 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 240 . 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 2579 . 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 2579 . 2  |-  ( E. y  e.  B  E. x  e.  A  ph  <->  E. y E. x ( ( y  e.  B  /\  x  e.  A )  /\  ph ) )
105, 7, 93bitr4i 268 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. y  e.  B  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   F/_wnfc 2406   E.wrex 2544
This theorem is referenced by:  rexcom  2701  rexcom4f  23134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549
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