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Theorem ralcomf 2834
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 1375 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  ->  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  ->  ph ) )
212albii 1573 . . 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 1748 . . 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 241 . 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 2709 . 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 2709 . 2  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
94, 6, 83bitr4i 269 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 177    /\ wa 359   A.wal 1546    e. wcel 1721   F/_wnfc 2535   A.wral 2674
This theorem is referenced by:  ralcom  2836  ssiinf  4108  ralcom4f  23926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679
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