Theorem bnj121 30940

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj121.1	|- ( ze <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
bnj121.2	|- ( ze' <-> [. 1o / n ]. ze )
bnj121.3	|- ( ph' <-> [. 1o / n ]. ph )
bnj121.4	|- ( ps' <-> [. 1o / n ]. ps )

Assertion

Ref	Expression
bnj121	|- ( ze' <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )

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

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

Proof of Theorem bnj121

Step	Hyp	Ref	Expression
1		bnj121.1	. . 3 |- ( ze <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
2	1	sbcbii 3491	. 2 |- ( [. 1o / n ]. ze <-> [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
3		bnj121.2	. 2 |- ( ze' <-> [. 1o / n ]. ze )
4		bnj105 30790	. . . . . . . 8 |- 1o e. _V
5	4	bnj90 30788	. . . . . . 7 |- ( [. 1o / n ]. f Fn n <-> f Fn 1o )
6	5	bicomi 214	. . . . . 6 |- ( f Fn 1o <-> [. 1o / n ]. f Fn n )
7		bnj121.3	. . . . . 6 |- ( ph' <-> [. 1o / n ]. ph )
8		bnj121.4	. . . . . 6 |- ( ps' <-> [. 1o / n ]. ps )
9	6, 7, 8	3anbi123i 1251	. . . . 5 |- ( ( f Fn 1o /\ ph' /\ ps' ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
10		sbc3an 3494	. . . . 5 |- ( [. 1o / n ]. ( f Fn n /\ ph /\ ps ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
11	9, 10	bitr4i 267	. . . 4 |- ( ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) )
12	11	imbi2i 326	. . 3 |- ( ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) )
13		nfv 1843	. . . . 5 |- F/ n ( R FrSe A /\ x e. A )
14	13	sbc19.21g 3502	. . . 4 |- ( 1o e. _V -> ( [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) ) )
15	4, 14	ax-mp 5	. . 3 |- ( [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) )
16	12, 15	bitr4i 267	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
17	2, 3, 16	3bitr4i 292	1 |- ( ze' <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   Fn wfn 5883   1oc1o 7553   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-pow 4843

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-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-suc 5729  df-fn 5891  df-1o 7560

This theorem is referenced by:  bnj150  30946  bnj153  30950