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Theorem bnj89 29023
 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 3202 . . 3
2 sbcal 3200 . . . 4
32exbii 1592 . . 3
4 bnj89.1 . . . . . . 7
5 sbcbig 3199 . . . . . . 7
64, 5ax-mp 8 . . . . . 6
7 sbcg 3218 . . . . . . . 8
84, 7ax-mp 8 . . . . . . 7
98bibi2i 305 . . . . . 6
106, 9bitri 241 . . . . 5
1110albii 1575 . . . 4
1211exbii 1592 . . 3
131, 3, 123bitri 263 . 2
14 df-eu 2284 . . 3
1514sbcbii 3208 . 2
16 df-eu 2284 . 2
1713, 15, 163bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wb 177  wal 1549  wex 1550   wcel 1725  weu 2280  cvv 2948  wsbc 3153 This theorem is referenced by:  bnj130  29182  bnj207  29189 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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