Theorem kmlem10 8981

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1	|- A = { u | E. t e. x u = ( t \ U. ( x \ { t } ) ) }

Assertion

Ref	Expression
kmlem10	|- ( A. h ( A. z e. h A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. h ph ) -> E. y A. z e. A ph )

Distinct variable groups:   x, y, z, w, u, t, h   y, A, z, w, h   ph, h

Allowed substitution hints:   ph( x, y, z, w, u, t)   A( x, u, t)

Proof of Theorem kmlem10

Step	Hyp	Ref	Expression
1		kmlem9.1	. . 3 |- A = { u | E. t e. x u = ( t \ U. ( x \ { t } ) ) }
2	1	kmlem9 8980	. 2 |- A. z e. A A. w e. A ( z =/= w -> ( z i^i w ) = (/) )
3		vex 3203	. . . . 5 |- x e. _V
4	3	abrexex 7141	. . . 4 |- { u | E. t e. x u = ( t \ U. ( x \ { t } ) ) } e. _V
5	1, 4	eqeltri 2697	. . 3 |- A e. _V
6		raleq 3138	. . . . 5 |- ( h = A -> ( A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) <-> A. w e. A ( z =/= w -> ( z i^i w ) = (/) ) ) )
7	6	raleqbi1dv 3146	. . . 4 |- ( h = A -> ( A. z e. h A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) <-> A. z e. A A. w e. A ( z =/= w -> ( z i^i w ) = (/) ) ) )
8		raleq 3138	. . . . 5 |- ( h = A -> ( A. z e. h ph <-> A. z e. A ph ) )
9	8	exbidv 1850	. . . 4 |- ( h = A -> ( E. y A. z e. h ph <-> E. y A. z e. A ph ) )
10	7, 9	imbi12d 334	. . 3 |- ( h = A -> ( ( A. z e. h A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. h ph ) <-> ( A. z e. A A. w e. A ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. A ph ) ) )
11	5, 10	spcv 3299	. 2 |- ( A. h ( A. z e. h A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. h ph ) -> ( A. z e. A A. w e. A ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. A ph ) )
12	2, 11	mpi 20	1 |- ( A. h ( A. z e. h A. w e. h ( z =/= w -> ( z i^i w ) = (/) ) -> E. y A. z e. h ph ) -> E. y A. z e. A ph )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   (/)c0 3915   {csn 4177   U.cuni 4436

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-rep 4771  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-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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  kmlem13  8984