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Theorem bnj89 29599
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj89.1
Assertion
Ref Expression
bnj89
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem bnj89
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcex2 3306 . . 3
2 sbcal 3305 . . . 4
32exbii 1726 . . 3
4 bnj89.1 . . . . . . 7
5 sbcbig 3300 . . . . . . 7
64, 5ax-mp 5 . . . . . 6
7 sbcg 3321 . . . . . . . 8
84, 7ax-mp 5 . . . . . . 7
98bibi2i 320 . . . . . 6
106, 9bitri 257 . . . . 5
1110albii 1699 . . . 4
1211exbii 1726 . . 3
131, 3, 123bitri 279 . 2
14 df-eu 2323 . . 3
1514sbcbii 3311 . 2
16 df-eu 2323 . 2
1713, 15, 163bitr4i 285 1
 Colors of variables: wff setvar class Syntax hints:   wb 189  wal 1450  wex 1671   wcel 1904  weu 2319  cvv 3031  wsbc 3255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256 This theorem is referenced by:  bnj130  29757  bnj207  29764
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