Theorem bnj556 30970

Description: Technical lemma for bnj852 30991. 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
bnj556.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj556.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )

Assertion

Ref	Expression
bnj556	|- ( et -> si )

Proof of Theorem bnj556

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- p e. _V
2	1	bnj216 30800	. . . 4 |- ( m = suc p -> p e. m )
3	2	3anim3i 1250	. . 3 |- ( ( m e. D /\ n = suc m /\ m = suc p ) -> ( m e. D /\ n = suc m /\ p e. m ) )
4	3	adantr 481	. 2 |- ( ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. om ) -> ( m e. D /\ n = suc m /\ p e. m ) )
5		bnj556.19	. . 3 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
6		bnj258 30774	. . 3 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. om ) )
7	5, 6	bitri 264	. 2 |- ( et <-> ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. om ) )
8		bnj556.18	. 2 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
9	4, 7, 8	3imtr4i 281	1 |- ( et -> si )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   suc csuc 5725   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729  df-bnj17 30753

This theorem is referenced by:  bnj557  30971  bnj561  30973  bnj562  30974