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Theorem ralcomf 2868
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 1379 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  ->  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  ->  ph ) )
212albii 1577 . . 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 1753 . . 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 242 . 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 2742 . 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 2742 . 2  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
94, 6, 83bitr4i 270 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 178    /\ wa 360   A.wal 1550    e. wcel 1726   F/_wnfc 2561   A.wral 2707
This theorem is referenced by:  ralcom  2870  ssiinf  4142  ralcom4f  23970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712
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