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Theorem ralcomf 3013
Description: Commutation of restricted universal 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 ancomst 450 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  ->  ph )  <->  ( (
y  e.  B  /\  x  e.  A )  ->  ph ) )
212albii 1646 . . 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 1850 . . 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 249 . 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 2830 . 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 2830 . 2  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. y A. x ( ( y  e.  B  /\  x  e.  A )  ->  ph )
)
94, 6, 83bitr4i 277 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. y  e.  B  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    e. wcel 1823   F/_wnfc 2602   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809
This theorem is referenced by:  ralcom  3015  ssiinf  4364  ralcom4f  27576
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