Theorem bnj986 31024

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj986.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj986.10	|- D = ( om \ { (/) } )
bnj986.15	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )

Assertion

Ref	Expression
bnj986	|- ( ch -> E. m E. p ta )

Distinct variable group:   m, n, p

Allowed substitution hints:   ph( f, m, n, p)   ps( f, m, n, p)   ch( f, m, n, p)   ta( f, m, n, p)   D( f, m, n, p)

Proof of Theorem bnj986

Step	Hyp	Ref	Expression
1		bnj986.3	. . . . . 6 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj986.10	. . . . . . 7 |- D = ( om \ { (/) } )
3	2	bnj158 30797	. . . . . 6 |- ( n e. D -> E. m e. om n = suc m )
4	1, 3	bnj769 30832	. . . . 5 |- ( ch -> E. m e. om n = suc m )
5	4	bnj1196 30865	. . . 4 |- ( ch -> E. m ( m e. om /\ n = suc m ) )
6		vex 3203	. . . . . 6 |- n e. _V
7	6	sucex 7011	. . . . 5 |- suc n e. _V
8	7	isseti 3209	. . . 4 |- E. p p = suc n
9	5, 8	jctir 561	. . 3 |- ( ch -> ( E. m ( m e. om /\ n = suc m ) /\ E. p p = suc n ) )
10		exdistr 1919	. . . 4 |- ( E. m E. p ( ( m e. om /\ n = suc m ) /\ p = suc n ) <-> E. m ( ( m e. om /\ n = suc m ) /\ E. p p = suc n ) )
11		19.41v 1914	. . . 4 |- ( E. m ( ( m e. om /\ n = suc m ) /\ E. p p = suc n ) <-> ( E. m ( m e. om /\ n = suc m ) /\ E. p p = suc n ) )
12	10, 11	bitr2i 265	. . 3 |- ( ( E. m ( m e. om /\ n = suc m ) /\ E. p p = suc n ) <-> E. m E. p ( ( m e. om /\ n = suc m ) /\ p = suc n ) )
13	9, 12	sylib 208	. 2 |- ( ch -> E. m E. p ( ( m e. om /\ n = suc m ) /\ p = suc n ) )
14		bnj986.15	. . . 4 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
15		df-3an 1039	. . . 4 |- ( ( m e. om /\ n = suc m /\ p = suc n ) <-> ( ( m e. om /\ n = suc m ) /\ p = suc n ) )
16	14, 15	bitri 264	. . 3 |- ( ta <-> ( ( m e. om /\ n = suc m ) /\ p = suc n ) )
17	16	2exbii 1775	. 2 |- ( E. m E. p ta <-> E. m E. p ( ( m e. om /\ n = suc m ) /\ p = suc n ) )
18	13, 17	sylibr 224	1 |- ( ch -> E. m E. p ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   suc csuc 5725   Fn wfn 5883   omcom 7065   /\ w-bnj17 30752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj996  31025