Theorem raltpg 4080

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

Hypotheses

Ref	Expression
ralprg.1
ralprg.2
raltpg.3

Assertion

Ref	Expression
raltpg

Distinct variable groups:   ,   ,   ,   ,   ,   ,

Proof of Theorem raltpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . 5
2		ralprg.2	. . . . 5
3	1, 2	ralprg 4078	. . . 4
4		raltpg.3	. . . . 5
5	4	ralsng 4064	. . . 4
6	3, 5	bi2anan9 873	. . 3
7	6	3impa 1191	. 2
8		df-tp 4034	. . . 4
9	8	raleqi 3058	. . 3
10		ralunb 3684	. . 3
11	9, 10	bitri 249	. 2
12		df-3an 975	. 2
13	7, 11, 12	3bitr4g 288	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  u.cun 3473  {csn 4029  {cpr 4031  {ctp 4033

This theorem is referenced by:  raltp  4084  raltpd  4153  f13dfv  6180  nb3grapr  24453  cusgra3v  24464  3v3e3cycl1  24644  constr3trllem2  24651  constr3trllem5  24654  frgra3v  25002

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  df-tp 4034