Theorem bnj124 30941

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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj124.1	|- F = { <. (/) , pred ( x , A , R ) >. }
bnj124.2	|- ( ph" <-> [. F / f ]. ph' )
bnj124.3	|- ( ps" <-> [. F / f ]. ps' )
bnj124.4	|- ( ze" <-> [. F / f ]. ze' )
bnj124.5	|- ( ze' <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )

Assertion

Ref	Expression
bnj124	|- ( ze" <-> ( ( R FrSe A /\ x e. A ) -> ( F Fn 1o /\ ph" /\ ps" ) ) )

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

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

Proof of Theorem bnj124

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj124.4	. 2 |- ( ze" <-> [. F / f ]. ze' )
2		bnj124.5	. . . 4 |- ( ze' <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )
3	2	sbcbii 3491	. . 3 |- ( [. F / f ]. ze' <-> [. F / f ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )
4		bnj124.1	. . . . 5 |- F = { <. (/) , pred ( x , A , R ) >. }
5	4	bnj95 30934	. . . 4 |- F e. _V
6		nfv 1843	. . . . 5 |- F/ f ( R FrSe A /\ x e. A )
7	6	sbc19.21g 3502	. . . 4 |- ( F e. _V -> ( [. F / f ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. F / f ]. ( f Fn 1o /\ ph' /\ ps' ) ) ) )
8	5, 7	ax-mp 5	. . 3 |- ( [. F / f ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. F / f ]. ( f Fn 1o /\ ph' /\ ps' ) ) )
9		fneq1 5979	. . . . . . . 8 |- ( f = z -> ( f Fn 1o <-> z Fn 1o ) )
10		fneq1 5979	. . . . . . . 8 |- ( z = F -> ( z Fn 1o <-> F Fn 1o ) )
11	9, 10	sbcie2g 3469	. . . . . . 7 |- ( F e. _V -> ( [. F / f ]. f Fn 1o <-> F Fn 1o ) )
12	5, 11	ax-mp 5	. . . . . 6 |- ( [. F / f ]. f Fn 1o <-> F Fn 1o )
13	12	bicomi 214	. . . . 5 |- ( F Fn 1o <-> [. F / f ]. f Fn 1o )
14		bnj124.2	. . . . 5 |- ( ph" <-> [. F / f ]. ph' )
15		bnj124.3	. . . . 5 |- ( ps" <-> [. F / f ]. ps' )
16	13, 14, 15, 5	bnj206 30799	. . . 4 |- ( [. F / f ]. ( f Fn 1o /\ ph' /\ ps' ) <-> ( F Fn 1o /\ ph" /\ ps" ) )
17	16	imbi2i 326	. . 3 |- ( ( ( R FrSe A /\ x e. A ) -> [. F / f ]. ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( F Fn 1o /\ ph" /\ ps" ) ) )
18	3, 8, 17	3bitri 286	. 2 |- ( [. F / f ]. ze' <-> ( ( R FrSe A /\ x e. A ) -> ( F Fn 1o /\ ph" /\ ps" ) ) )
19	1, 18	bitri 264	1 |- ( ze" <-> ( ( R FrSe A /\ x e. A ) -> ( F Fn 1o /\ ph" /\ ps" ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   (/)c0 3915   {csn 4177   <.cop 4183   Fn wfn 5883   1oc1o 7553   predc-bnj14 30754   FrSe w-bnj15 30758

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-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-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  bnj150  30946  bnj153  30950