Theorem bnj207 30951

Description: Technical lemma for bnj852 30991. 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
bnj207.1	|- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj207.2	|- ( ph' <-> [. M / n ]. ph )
bnj207.3	|- ( ps' <-> [. M / n ]. ps )
bnj207.4	|- ( ch' <-> [. M / n ]. ch )
bnj207.5	|- M e. _V

Assertion

Ref	Expression
bnj207	|- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) )

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

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

Proof of Theorem bnj207

Step	Hyp	Ref	Expression
1		bnj207.4	. 2 |- ( ch' <-> [. M / n ]. ch )
2		bnj207.1	. . . 4 |- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
3	2	sbcbii 3491	. . 3 |- ( [. M / n ]. ch <-> [. M / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
4		bnj207.5	. . . . 5 |- M e. _V
5		nfv 1843	. . . . . 6 |- F/ n ( R FrSe A /\ x e. A )
6	5	sbc19.21g 3502	. . . . 5 |- ( M e. _V -> ( [. M / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) ) )
7	4, 6	ax-mp 5	. . . 4 |- ( [. M / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) )
8	4	bnj89 30787	. . . . . 6 |- ( [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) <-> E! f [. M / n ]. ( f Fn n /\ ph /\ ps ) )
9	4	bnj90 30788	. . . . . . . . 9 |- ( [. M / n ]. f Fn n <-> f Fn M )
10	9	bicomi 214	. . . . . . . 8 |- ( f Fn M <-> [. M / n ]. f Fn n )
11		bnj207.2	. . . . . . . 8 |- ( ph' <-> [. M / n ]. ph )
12		bnj207.3	. . . . . . . 8 |- ( ps' <-> [. M / n ]. ps )
13	10, 11, 12, 4	bnj206 30799	. . . . . . 7 |- ( [. M / n ]. ( f Fn n /\ ph /\ ps ) <-> ( f Fn M /\ ph' /\ ps' ) )
14	13	eubii 2492	. . . . . 6 |- ( E! f [. M / n ]. ( f Fn n /\ ph /\ ps ) <-> E! f ( f Fn M /\ ph' /\ ps' ) )
15	8, 14	bitri 264	. . . . 5 |- ( [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) <-> E! f ( f Fn M /\ ph' /\ ps' ) )
16	15	imbi2i 326	. . . 4 |- ( ( ( R FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) )
17	7, 16	bitri 264	. . 3 |- ( [. M / n ]. ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) )
18	3, 17	bitri 264	. 2 |- ( [. M / n ]. ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) )
19	1, 18	bitri 264	1 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.wsbc 3435   Fn wfn 5883   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

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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-fn 5891

This theorem is referenced by:  bnj600  30989  bnj908  31001