Theorem bnj1049 31042

Description: Technical lemma for bnj69 31078. 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
bnj1049.1	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1049.2	|- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )

Assertion

Ref	Expression
bnj1049	|- ( A. i e. n et <-> A. i et )

Proof of Theorem bnj1049

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. i e. n et <-> A. i ( i e. n -> et ) )
2		bnj1049.2	. . . . . . 7 |- ( et <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
3	2	imbi2i 326	. . . . . 6 |- ( ( i e. n -> et ) <-> ( i e. n -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) ) )
4		impexp 462	. . . . . 6 |- ( ( ( i e. n /\ ( th /\ ta /\ ch /\ ze ) ) -> z e. B ) <-> ( i e. n -> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) ) )
5	3, 4	bitr4i 267	. . . . 5 |- ( ( i e. n -> et ) <-> ( ( i e. n /\ ( th /\ ta /\ ch /\ ze ) ) -> z e. B ) )
6		bnj1049.1	. . . . . . . . . 10 |- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
7	6	simplbi 476	. . . . . . . . 9 |- ( ze -> i e. n )
8	7	bnj708 30826	. . . . . . . 8 |- ( ( th /\ ta /\ ch /\ ze ) -> i e. n )
9	8	pm4.71ri 665	. . . . . . 7 |- ( ( th /\ ta /\ ch /\ ze ) <-> ( i e. n /\ ( th /\ ta /\ ch /\ ze ) ) )
10	9	bicomi 214	. . . . . 6 |- ( ( i e. n /\ ( th /\ ta /\ ch /\ ze ) ) <-> ( th /\ ta /\ ch /\ ze ) )
11	10	imbi1i 339	. . . . 5 |- ( ( ( i e. n /\ ( th /\ ta /\ ch /\ ze ) ) -> z e. B ) <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
12	5, 11	bitri 264	. . . 4 |- ( ( i e. n -> et ) <-> ( ( th /\ ta /\ ch /\ ze ) -> z e. B ) )
13	12, 2	bitr4i 267	. . 3 |- ( ( i e. n -> et ) <-> et )
14	13	albii 1747	. 2 |- ( A. i ( i e. n -> et ) <-> A. i et )
15	1, 14	bitri 264	1 |- ( A. i e. n et <-> A. i et )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912   ` cfv 5888   /\ w-bnj17 30752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917  df-bnj17 30753

This theorem is referenced by:  bnj1052  31043