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Theorem bnj1533 33643
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 33589 . . 3  |-  ( th 
->  A. z ( z  e.  B  ->  -.  z  e.  D )
)
3 bnj1533.3 . . . . . . . . 9  |-  D  =  { z  e.  A  |  C  =/=  E }
43rabeq2i 3092 . . . . . . . 8  |-  ( z  e.  D  <->  ( z  e.  A  /\  C  =/= 
E ) )
54notbii 296 . . . . . . 7  |-  ( -.  z  e.  D  <->  -.  (
z  e.  A  /\  C  =/=  E ) )
6 imnan 422 . . . . . . 7  |-  ( ( z  e.  A  ->  -.  C  =/=  E
)  <->  -.  ( z  e.  A  /\  C  =/= 
E ) )
7 nne 2644 . . . . . . . 8  |-  ( -.  C  =/=  E  <->  C  =  E )
87imbi2i 312 . . . . . . 7  |-  ( ( z  e.  A  ->  -.  C  =/=  E
)  <->  ( z  e.  A  ->  C  =  E ) )
95, 6, 83bitr2i 273 . . . . . 6  |-  ( -.  z  e.  D  <->  ( z  e.  A  ->  C  =  E ) )
109imbi2i 312 . . . . 5  |-  ( ( z  e.  B  ->  -.  z  e.  D
)  <->  ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) ) )
11 bnj1533.2 . . . . . . . 8  |-  B  C_  A
1211sseli 3485 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  A )
1312imim1i 58 . . . . . 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 568 . . . . . . . . 9  |-  ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B
)  ->  ( (
z  e.  B  -> 
( z  e.  A  ->  C  =  E ) )  /\  z  e.  B ) )
16 simpr 461 . . . . . . . . . . 11  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  z  e.  B )
17 simpl 457 . . . . . . . . . . 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 532 . . . . . . . . 9  |-  ( ( ( z  e.  B  ->  ( z  e.  A  ->  C  =  E ) )  /\  z  e.  B )  ->  (
( z  e.  A  ->  C  =  E )  /\  z  e.  B
) )
2015, 19impbii 188 . . . . . . . 8  |-  ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B
)  <->  ( ( z  e.  B  ->  (
z  e.  A  ->  C  =  E )
)  /\  z  e.  B ) )
2120imbi1i 325 . . . . . . 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 446 . . . . . . 7  |-  ( ( ( ( z  e.  A  ->  C  =  E )  /\  z  e.  B )  ->  C  =  E )  <->  ( (
z  e.  A  ->  C  =  E )  ->  ( z  e.  B  ->  C  =  E ) ) )
23 impexp 446 . . . . . . 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 275 . . . . . 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 208 . . . . 5  |-  ( ( z  e.  B  -> 
( z  e.  A  ->  C  =  E ) )  ->  ( z  e.  B  ->  C  =  E ) )
2610, 25sylbi 195 . . . 4  |-  ( ( z  e.  B  ->  -.  z  e.  D
)  ->  ( z  e.  B  ->  C  =  E ) )
2726alimi 1620 . . 3  |-  ( A. z ( z  e.  B  ->  -.  z  e.  D )  ->  A. z
( z  e.  B  ->  C  =  E ) )
282, 27syl 16 . 2  |-  ( th 
->  A. z ( z  e.  B  ->  C  =  E ) )
2928bnj1142 33581 1  |-  ( th 
->  A. z  e.  B  C  =  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-ne 2640  df-ral 2798  df-rab 2802  df-in 3468  df-ss 3475
This theorem is referenced by:  bnj1523  33860
  Copyright terms: Public domain W3C validator