Theorem bnj1171 31068

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
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 3602	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( w e. B -> w e. A ) )
4	3	pm4.71rd 667	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( w e. B <-> ( w e. A /\ w e. B ) ) )
5	4	imbi1d 331	. . . . . . . 8 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( ( w e. A /\ w e. B ) -> -. w R z ) ) )
6		impexp 462	. . . . . . . 8 |- ( ( ( w e. A /\ w e. B ) -> -. w R z ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) )
7	5, 6	syl6bb 276	. . . . . . 7 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) ) )
8		con2b 349	. . . . . . . 8 |- ( ( w R z -> -. w e. B ) <-> ( w e. B -> -. w R z ) )
9	8	imbi2i 326	. . . . . . 7 |- ( ( w e. A -> ( w R z -> -. w e. B ) ) <-> ( w e. A -> ( w e. B -> -. w R z ) ) )
10	7, 9	syl6bbr 278	. . . . . 6 |- ( ( ph /\ ps ) -> ( ( w e. B -> -. w R z ) <-> ( w e. A -> ( w R z -> -. w e. B ) ) ) )
11	10	anbi2d 740	. . . . 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 260	. . . 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 1747	. . 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 1774	. 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 221	1 |- E. z A. w ( ( ph /\ ps ) -> ( z e. B /\ ( w e. B -> -. w R z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   C_ wss 3574   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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  bnj1190  31076