Theorem bnj125 30942

Description: Technical lemma for bnj150 30946. 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
bnj125.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj125.2	|- ( ph' <-> [. 1o / n ]. ph )
bnj125.3	|- ( ph" <-> [. F / f ]. ph' )
bnj125.4	|- F = { <. (/) , pred ( x , A , R ) >. }

Assertion

Ref	Expression
bnj125	|- ( ph" <-> ( F ` (/) ) = pred ( x , A , R ) )

Distinct variable groups:   A, f, n   f, F   R, f, n   x, f, n

Allowed substitution hints:   ph( x, f, n)   A( x)   R( x)   F( x, n)   ph'( x, f, n)   ph"( x, f, n)

Proof of Theorem bnj125

Step	Hyp	Ref	Expression
1		bnj125.3	. 2 |- ( ph" <-> [. F / f ]. ph' )
2		bnj125.2	. . . 4 |- ( ph' <-> [. 1o / n ]. ph )
3	2	sbcbii 3491	. . 3 |- ( [. F / f ]. ph' <-> [. F / f ]. [. 1o / n ]. ph )
4		bnj125.1	. . . . . 6 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
5		bnj105 30790	. . . . . 6 |- 1o e. _V
6	4, 5	bnj91 30931	. . . . 5 |- ( [. 1o / n ]. ph <-> ( f ` (/) ) = pred ( x , A , R ) )
7	6	sbcbii 3491	. . . 4 |- ( [. F / f ]. [. 1o / n ]. ph <-> [. F / f ]. ( f ` (/) ) = pred ( x , A , R ) )
8		bnj125.4	. . . . . 6 |- F = { <. (/) , pred ( x , A , R ) >. }
9	8	bnj95 30934	. . . . 5 |- F e. _V
10		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` (/) ) = ( F ` (/) ) )
11	10	eqeq1d 2624	. . . . 5 |- ( f = F -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ( F ` (/) ) = pred ( x , A , R ) ) )
12	9, 11	sbcie 3470	. . . 4 |- ( [. F / f ]. ( f ` (/) ) = pred ( x , A , R ) <-> ( F ` (/) ) = pred ( x , A , R ) )
13	7, 12	bitri 264	. . 3 |- ( [. F / f ]. [. 1o / n ]. ph <-> ( F ` (/) ) = pred ( x , A , R ) )
14	3, 13	bitri 264	. 2 |- ( [. F / f ]. ph' <-> ( F ` (/) ) = pred ( x , A , R ) )
15	1, 14	bitri 264	1 |- ( ph" <-> ( F ` (/) ) = pred ( x , A , R ) )

Syntax hints:   <-> wb 196   = wceq 1483   [.wsbc 3435   (/)c0 3915   {csn 4177   <.cop 4183   ` cfv 5888   1oc1o 7553   predc-bnj14 30754

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  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-br 4654  df-suc 5729  df-iota 5851  df-fv 5896  df-1o 7560

This theorem is referenced by:  bnj150  30946  bnj153  30950