MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r2exf Structured version   Unicode version

Theorem r2exf 2927
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 2926. (Revised by Wolf Lammen, 10-Jan-2020.)
Hypothesis
Ref Expression
r2exf.1  |-  F/_ y A
Assertion
Ref Expression
r2exf  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem r2exf
StepHypRef Expression
1 r2exf.1 . . 3  |-  F/_ y A
21r2alf 2779 . 2  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  A. x A. y ( ( x  e.  A  /\  y  e.  B
)  ->  -.  ph )
)
32r2exlem 2926 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367   E.wex 1633    e. wcel 1842   F/_wnfc 2550   E.wrex 2754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759
This theorem is referenced by:  r2exOLD  2930  rexcomf  2966
  Copyright terms: Public domain W3C validator