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Theorem bnj1533 29615
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1533.1  |-  ( th 
->  A. z  e.  B  -.  z  e.  D
)
bnj1533.2  |-  B  C_  A
bnj1533.3  |-  D  =  { z  e.  A  |  C  =/=  E }
Assertion
Ref Expression
bnj1533  |-  ( th 
->  A. z  e.  B  C  =  E )

Proof of Theorem bnj1533
StepHypRef Expression
1 bnj1533.1 . . . 4  |-  ( th 
->  A. z  e.  B  -.  z  e.  D
)
21bnj1211 29561 . . 3  |-  ( th 
->  A. z ( z  e.  B  ->  -.  z  e.  D )
)
3 bnj1533.3 . . . . . . . . 9  |-  D  =  { z  e.  A  |  C  =/=  E }
43rabeq2i 3019 . . . . . . . 8  |-  ( z  e.  D  <->  ( z  e.  A  /\  C  =/= 
E ) )
54notbii 297 . . . . . . 7  |-  ( -.  z  e.  D  <->  -.  (
z  e.  A  /\  C  =/=  E ) )
6 imnan 423 . . . . . . 7  |-  ( ( z  e.  A  ->  -.  C  =/=  E
)  <->  -.  ( z  e.  A  /\  C  =/= 
E ) )
7 nne 2605 . . . . . . . 8  |-  ( -.  C  =/=  E  <->  C  =  E )
87imbi2i 313 . . . . . . 7  |-  ( ( z  e.  A  ->  -.  C  =/=  E
)  <->  ( z  e.  A  ->  C  =  E ) )
95, 6, 83bitr2i 276 . . . . . 6  |-  ( -.  z  e.  D  <->  ( z  e.  A  ->  C  =  E ) )
109imbi2i 313 . . . . 5  |-  ( ( z  e.  B  ->  -.  z  e.  D
)  <->  ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) ) )
11 bnj1533.2 . . . . . . . 8  |-  B  C_  A
1211sseli 3403 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  A )
1312imim1i 60 . . . . . 6  |-  ( ( z  e.  A  ->  C  =  E )  ->  ( z  e.  B  ->  C  =  E ) )
14 ax-1 6 . . . . . . . . . 10  |-  ( ( z  e.  A  ->  C  =  E )  ->  ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) ) )
1514anim1i 570 . . . . . . . . 9  |-  ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B
)  ->  ( (
z  e.  B  -> 
( z  e.  A  ->  C  =  E ) )  /\  z  e.  B ) )
16 simpr 462 . . . . . . . . . . 11  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  z  e.  B )
17 simpl 458 . . . . . . . . . . 11  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  (
z  e.  B  -> 
( z  e.  A  ->  C  =  E ) ) )
1816, 17mpd 15 . . . . . . . . . 10  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  (
z  e.  A  ->  C  =  E )
)
1918, 16jca 534 . . . . . . . . 9  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  (
( z  e.  A  ->  C  =  E )  /\  z  e.  B
) )
2015, 19impbii 190 . . . . . . . 8  |-  ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B
)  <->  ( ( z  e.  B  ->  (
z  e.  A  ->  C  =  E )
)  /\  z  e.  B ) )
2120imbi1i 326 . . . . . . 7  |-  ( ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B )  ->  C  =  E )  <->  ( (
( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  C  =  E ) )
22 impexp 447 . . . . . . 7  |-  ( ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B )  ->  C  =  E )  <->  ( (
z  e.  A  ->  C  =  E )  ->  ( z  e.  B  ->  C  =  E ) ) )
23 impexp 447 . . . . . . 7  |-  ( ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  C  =  E )  <-> 
( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  -> 
( z  e.  B  ->  C  =  E ) ) )
2421, 22, 233bitr3i 278 . . . . . 6  |-  ( ( ( z  e.  A  ->  C  =  E )  ->  ( z  e.  B  ->  C  =  E ) )  <->  ( (
z  e.  B  -> 
( z  e.  A  ->  C  =  E ) )  ->  ( z  e.  B  ->  C  =  E ) ) )
2513, 24mpbi 211 . . . . 5  |-  ( ( z  e.  B  -> 
( z  e.  A  ->  C  =  E ) )  ->  ( z  e.  B  ->  C  =  E ) )
2610, 25sylbi 198 . . . 4  |-  ( ( z  e.  B  ->  -.  z  e.  D
)  ->  ( z  e.  B  ->  C  =  E ) )
2726alimi 1678 . . 3  |-  ( A. z ( z  e.  B  ->  -.  z  e.  D )  ->  A. z
( z  e.  B  ->  C  =  E ) )
282, 27syl 17 . 2  |-  ( th 
->  A. z ( z  e.  B  ->  C  =  E ) )
2928bnj1142 29553 1  |-  ( th 
->  A. z  e.  B  C  =  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   {crab 2718    C_ wss 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-ne 2601  df-ral 2719  df-rab 2723  df-in 3386  df-ss 3393
This theorem is referenced by:  bnj1523  29832
  Copyright terms: Public domain W3C validator