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Theorem ralcomf 2698
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
ralcomf  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem ralcomf
StepHypRef Expression
1 ancomsimp 1359 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  ->  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  ->  ph ) )
212albii 1554 . . 3  |-  ( A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )  <->  A. x A. y ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
3 alcom 1711 . . 3  |-  ( A. x A. y ( ( y  e.  B  /\  x  e.  A )  ->  ph )  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
42, 3bitri 240 . 2  |-  ( A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
5 ralcomf.1 . . 3  |-  F/_ y A
65r2alf 2578 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )
)
7 ralcomf.2 . . 3  |-  F/_ x B
87r2alf 2578 . 2  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
94, 6, 83bitr4i 268 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684   F/_wnfc 2406   A.wral 2543
This theorem is referenced by:  ralcom  2700  ssiinf  3951  ralcom4f  23133
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-ral 2548
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