Theorem onfrALTlem4 38758

Description: Lemma for onfrALT 38764. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem4	|- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )

Distinct variable group:   x, a

Proof of Theorem onfrALTlem4

Step	Hyp	Ref	Expression
1		sbcan 3478	. 2 |- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( [. y / x ]. x e. a /\ [. y / x ]. ( a i^i x ) = (/) ) )
2		sbcel1v 3495	. . 3 |- ( [. y / x ]. x e. a <-> y e. a )
3		vex 3203	. . . . 5 |- y e. _V
4		sbceqg 3984	. . . . 5 |- ( y e. _V -> ( [. y / x ]. ( a i^i x ) = (/) <-> [_ y / x ]_ ( a i^i x ) = [_ y / x ]_ (/) ) )
5	3, 4	ax-mp 5	. . . 4 |- ( [. y / x ]. ( a i^i x ) = (/) <-> [_ y / x ]_ ( a i^i x ) = [_ y / x ]_ (/) )
6		csbin 4010	. . . . . 6 |- [_ y / x ]_ ( a i^i x ) = ( [_ y / x ]_ a i^i [_ y / x ]_ x )
7		csbconstg 3546	. . . . . . . 8 |- ( y e. _V -> [_ y / x ]_ a = a )
8	3, 7	ax-mp 5	. . . . . . 7 |- [_ y / x ]_ a = a
9		csbvarg 4003	. . . . . . . 8 |- ( y e. _V -> [_ y / x ]_ x = y )
10	3, 9	ax-mp 5	. . . . . . 7 |- [_ y / x ]_ x = y
11	8, 10	ineq12i 3812	. . . . . 6 |- ( [_ y / x ]_ a i^i [_ y / x ]_ x ) = ( a i^i y )
12	6, 11	eqtri 2644	. . . . 5 |- [_ y / x ]_ ( a i^i x ) = ( a i^i y )
13		csb0 3982	. . . . 5 |- [_ y / x ]_ (/) = (/)
14	12, 13	eqeq12i 2636	. . . 4 |- ( [_ y / x ]_ ( a i^i x ) = [_ y / x ]_ (/) <-> ( a i^i y ) = (/) )
15	5, 14	bitri 264	. . 3 |- ( [. y / x ]. ( a i^i x ) = (/) <-> ( a i^i y ) = (/) )
16	2, 15	anbi12i 733	. 2 |- ( ( [. y / x ]. x e. a /\ [. y / x ]. ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
17	1, 16	bitri 264	1 |- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   i^i cin 3573   (/)c0 3915

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-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  onfrALTlem1  38763  onfrALTlem1VD  39126