Theorem rexprg 4079

Description: Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralprg.1
ralprg.2

Assertion

Ref	Expression
rexprg

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem rexprg

Step	Hyp	Ref	Expression
1		df-pr 4032	. . . 4
2	1	rexeqi 3059	. . 3
3		rexun 3683	. . 3
4	2, 3	bitri 249	. 2
5		ralprg.1	. . . . 5
6	5	rexsng 4065	. . . 4
7	6	orbi1d 702	. . 3
8		ralprg.2	. . . . 5
9	8	rexsng 4065	. . . 4
10	9	orbi2d 701	. . 3
11	7, 10	sylan9bb 699	. 2
12	4, 11	syl5bb 257	1

Syntax hints:  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808  u.cun 3473  {csn 4029  {cpr 4031

This theorem is referenced by:  rextpg  4081  rexpr  4083  fr2nr  4862  sgrp2nmndlem5  16047  nb3graprlem2  24452  frgra2v  24999  3vfriswmgralem  25004  ldepspr  33074  zlmodzxzldeplem4  33104

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-rex 2813  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032