Theorem ralpr 4082

Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralpr.1
ralpr.2
ralpr.3
ralpr.4

Assertion

Ref	Expression
ralpr

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem ralpr

Step	Hyp	Ref	Expression
1		ralpr.1	. 2
2		ralpr.2	. 2
3		ralpr.3	. . 3
4		ralpr.4	. . 3
5	3, 4	ralprg 4078	. 2
6	1, 2, 5	mp2an 672	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  cvv 3109  {cpr 4031

This theorem is referenced by:  fzprval  11769  wwlktovf1  12895  xpsfrnel  14960  xpsle  14978  isdrs2  15568  pmtrsn  16544  iblcnlem1  22194  wlkntrllem2  24562  wlkntrllem3  24563  2wlklem  24566  numclwwlkovf2ex  25086  subfacp1lem3  28626  fprb  29203  ldepsnlinc  33109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032