MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.42vv Unicode version

Theorem 19.42vv 1850
Description: Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
Assertion
Ref Expression
19.42vv  |-  ( E. x E. y (
ph  /\  ps )  <->  (
ph  /\  E. x E. y ps ) )
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    ps( x, y)

Proof of Theorem 19.42vv
StepHypRef Expression
1 exdistr 1849 . 2  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
2 19.42v 1848 . 2  |-  ( E. x ( ph  /\  E. y ps )  <->  ( ph  /\ 
E. x E. y ps ) )
31, 2bitri 240 1  |-  ( E. x E. y (
ph  /\  ps )  <->  (
ph  /\  E. x E. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1530
This theorem is referenced by:  19.42vvv  1851  exdistr2  1852  3exdistr  1853  ceqsex3v  2828  ceqsex4v  2829  ceqsex8v  2831  elvvv  4751  dfoprab2  5897  resoprab  5942  oprabex3  5964  ov3  5986  ov6g  5987  xpassen  6958  axaddf  8769  axmulf  8770  brimg  24478  bnj996  29060  dvhopellsm  31380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-11 1717
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1531  df-nf 1534
  Copyright terms: Public domain W3C validator