Theorem bnj1171 33153

Description: Technical lemma for bnj69 33163. 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
bnj1171.13	|- ( ( ph /\ ps ) -> B C_ A )
bnj1171.129	|- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )

Assertion

Ref	Expression
bnj1171	|- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) )

Proof of Theorem bnj1171

Step	Hyp	Ref	Expression
1		bnj1171.129	. 2 |- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) )
2		bnj1171.13	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> B C_ A )
3	2	sseld 3503	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( w e. B -> w e. A ) )
4	3	pm4.71rd 635	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( w e. B <-> ( w e. A /\ w e. B ) ) )
5	4	imbi1d 317	. . . . . . . 8 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( ( w e. A /\ w e. B ) -> -. w R z ) ) )
6		impexp 446	. . . . . . . 8 |- ( ( ( w e. A /\ w e. B ) -> -. w R z ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) )
7	5, 6	syl6bb 261	. . . . . . 7 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) ) )
8		con2b 334	. . . . . . . 8 |- ( ( w R z -> -. w e. B ) <-> ( w e. B -> -. w R z ) )
9	8	imbi2i 312	. . . . . . 7 |- ( ( w e. A -> ( w R z -> -. w e. B ) ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) )
10	7, 9	syl6bbr 263	. . . . . 6 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( w e. A -> ( w R z -> -. w e. B ) ) ) )
11	10	anbi2d 703	. . . . 5 |- ( ( ph /\ ps ) -> ( ( z e. B /\ ( w e. B -> -. w R z ) ) <-> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
12	11	pm5.74i 245	. . . 4 |- ( ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) ) <-> ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
13	12	albii 1620	. . 3 |- ( A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) ) <-> A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
14	13	exbii 1644	. 2 |- ( E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) ) <-> E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. A -> ( w R z -> -. w e. B ) ) ) ) )
15	1, 14	mpbir 209	1 |- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   A.wal 1377   E.wex 1596   e. wcel 1767   C_ wss 3476   class class class wbr 4447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490

This theorem is referenced by:  bnj1190  33161