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Theorem bnj1049 31852
Description: Technical lemma for bnj69 31888. 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
StepHypRef Expression
1 df-ral 2715 . 2  |-  ( A. i  e.  n  et  <->  A. i ( i  e.  n  ->  et )
)
2 bnj1049.2 . . . . . . 7  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
32imbi2i 312 . . . . . 6  |-  ( ( i  e.  n  ->  et )  <->  ( i  e.  n  ->  ( ( th  /\  ta  /\  ch  /\ 
ze )  ->  z  e.  B ) ) )
4 impexp 446 . . . . . 6  |-  ( ( ( i  e.  n  /\  ( th  /\  ta  /\ 
ch  /\  ze )
)  ->  z  e.  B )  <->  ( i  e.  n  ->  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) ) )
53, 4bitr4i 252 . . . . 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 ) ) )
76simplbi 460 . . . . . . . . 9  |-  ( ze 
->  i  e.  n
)
87bnj708 31635 . . . . . . . 8  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
i  e.  n )
98pm4.71ri 633 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch  /\  ze )  <->  ( i  e.  n  /\  ( th  /\  ta  /\  ch  /\ 
ze ) ) )
109bicomi 202 . . . . . 6  |-  ( ( i  e.  n  /\  ( th  /\  ta  /\  ch  /\  ze ) )  <-> 
( th  /\  ta  /\ 
ch  /\  ze )
)
1110imbi1i 325 . . . . 5  |-  ( ( ( i  e.  n  /\  ( th  /\  ta  /\ 
ch  /\  ze )
)  ->  z  e.  B )  <->  ( ( th  /\  ta  /\  ch  /\ 
ze )  ->  z  e.  B ) )
125, 11bitri 249 . . . 4  |-  ( ( i  e.  n  ->  et )  <->  ( ( th 
/\  ta  /\  ch  /\  ze )  ->  z  e.  B ) )
1312, 2bitr4i 252 . . 3  |-  ( ( i  e.  n  ->  et )  <->  et )
1413albii 1610 . 2  |-  ( A. i ( i  e.  n  ->  et )  <->  A. i et )
151, 14bitri 249 1  |-  ( A. i  e.  n  et  <->  A. i et )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    e. wcel 1756   A.wral 2710   ` cfv 5413    /\ w-bnj17 31561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-an 371  df-ral 2715  df-bnj17 31562
This theorem is referenced by:  bnj1052  31853
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