Theorem lhpexle1lem 35293

Description: Lemma for lhpexle1 35294 and others that eliminates restrictions on X. (Contributed by NM, 24-Jul-2013.)

Hypotheses

Ref	Expression
lhpexle1lem.1	|- ( ph -> E. p e. A ( p .<_ W /\ ps ) )
lhpexle1lem.2	|- ( ( ph /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )

Assertion

Ref	Expression
lhpexle1lem	|- ( ph -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )

Distinct variable groups:   .<_ , p   A, p   W, p   X, p   ph, p

Allowed substitution hint:   ps( p)

Proof of Theorem lhpexle1lem

Step	Hyp	Ref	Expression
1		lhpexle1lem.1	. . . 4 |- ( ph -> E. p e. A ( p .<_ W /\ ps ) )
2	1	adantr 481	. . 3 |- ( ( ph /\ -. X e. A ) -> E. p e. A ( p .<_ W /\ ps ) )
3		simprl 794	. . . . . 6 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> p .<_ W )
4		simprr 796	. . . . . 6 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> ps )
5		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> p e. A )
6		simpllr 799	. . . . . . 7 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> -. X e. A )
7		nelne2 2891	. . . . . . 7 |- ( ( p e. A /\ -. X e. A ) -> p =/= X )
8	5, 6, 7	syl2anc 693	. . . . . 6 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> p =/= X )
9	3, 4, 8	3jca 1242	. . . . 5 |- ( ( ( ( ph /\ -. X e. A ) /\ p e. A ) /\ ( p .<_ W /\ ps ) ) -> ( p .<_ W /\ ps /\ p =/= X ) )
10	9	ex 450	. . . 4 |- ( ( ( ph /\ -. X e. A ) /\ p e. A ) -> ( ( p .<_ W /\ ps ) -> ( p .<_ W /\ ps /\ p =/= X ) ) )
11	10	reximdva 3017	. . 3 |- ( ( ph /\ -. X e. A ) -> ( E. p e. A ( p .<_ W /\ ps ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) ) )
12	2, 11	mpd 15	. 2 |- ( ( ph /\ -. X e. A ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )
13	1	adantr 481	. . 3 |- ( ( ph /\ -. X .<_ W ) -> E. p e. A ( p .<_ W /\ ps ) )
14		simprl 794	. . . . . 6 |- ( ( ( ph /\ -. X .<_ W ) /\ ( p .<_ W /\ ps ) ) -> p .<_ W )
15		simprr 796	. . . . . 6 |- ( ( ( ph /\ -. X .<_ W ) /\ ( p .<_ W /\ ps ) ) -> ps )
16		simplr 792	. . . . . . 7 |- ( ( ( ph /\ -. X .<_ W ) /\ ( p .<_ W /\ ps ) ) -> -. X .<_ W )
17		nbrne2 4673	. . . . . . 7 |- ( ( p .<_ W /\ -. X .<_ W ) -> p =/= X )
18	14, 16, 17	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ -. X .<_ W ) /\ ( p .<_ W /\ ps ) ) -> p =/= X )
19	14, 15, 18	3jca 1242	. . . . 5 |- ( ( ( ph /\ -. X .<_ W ) /\ ( p .<_ W /\ ps ) ) -> ( p .<_ W /\ ps /\ p =/= X ) )
20	19	ex 450	. . . 4 |- ( ( ph /\ -. X .<_ W ) -> ( ( p .<_ W /\ ps ) -> ( p .<_ W /\ ps /\ p =/= X ) ) )
21	20	reximdv 3016	. . 3 |- ( ( ph /\ -. X .<_ W ) -> ( E. p e. A ( p .<_ W /\ ps ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) ) )
22	13, 21	mpd 15	. 2 |- ( ( ph /\ -. X .<_ W ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )
23		lhpexle1lem.2	. 2 |- ( ( ph /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )
24	12, 22, 23	pm2.61dda 834	1 |- ( ph -> E. p e. A ( p .<_ W /\ ps /\ p =/= X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  lhpexle1  35294  lhpexle2  35296  lhpexle3  35298