Theorem onfrALTlem1 38763

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

Assertion

Ref	Expression
onfrALTlem1	|- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )

Distinct variable group:   x, a, y

Proof of Theorem onfrALTlem1

Step	Hyp	Ref	Expression
1		19.8a 2052	. . . . 5 |- ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. x ( x e. a /\ ( a i^i x ) = (/) ) )
2	1	a1i 11	. . . 4 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. x ( x e. a /\ ( a i^i x ) = (/) ) ) )
3		cbvexsv 38762	. . . 4 |- ( E. x ( x e. a /\ ( a i^i x ) = (/) ) <-> E. y [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) )
4	2, 3	syl6ib 241	. . 3 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. y [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) ) )
5		sbsbc 3439	. . . . 5 |- ( [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) )
6		onfrALTlem4 38758	. . . . 5 |- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
7	5, 6	bitri 264	. . . 4 |- ( [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
8	7	exbii 1774	. . 3 |- ( E. y [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
9	4, 8	syl6ib 241	. 2 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. y ( y e. a /\ ( a i^i y ) = (/) ) ) )
10		df-rex 2918	. 2 |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
11	9, 10	syl6ibr 242	1 |- ( ( a C_ On /\ a =/= (/) ) -> ( ( x e. a /\ ( a i^i x ) = (/) ) -> E. y e. a ( a i^i y ) = (/) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   =/= wne 2794   E.wrex 2913   [.wsbc 3435   i^i cin 3573   C_ wss 3574   (/)c0 3915   Oncon0 5723

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-ral 2917  df-rex 2918  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:  onfrALT  38764