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Theorem bnj23 29083
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj23.1
Assertion
Ref Expression
bnj23
Distinct variable groups:   ,   ,,   ,,,   ,,,
Allowed substitution hints:   (,,,)   ()   ()   ()

Proof of Theorem bnj23
StepHypRef Expression
1 vex 2959 . . . . 5
2 sbcng 3201 . . . . 5
31, 2ax-mp 8 . . . 4
4 bnj23.1 . . . . . . . 8
54eleq2i 2500 . . . . . . 7
6 nfcv 2572 . . . . . . . 8
76elrabsf 3199 . . . . . . 7
85, 7bitri 241 . . . . . 6
9 breq1 4215 . . . . . . . 8
109notbid 286 . . . . . . 7
1110rspccv 3049 . . . . . 6
128, 11syl5bir 210 . . . . 5
1312expdimp 427 . . . 4
143, 13syl5bir 210 . . 3
1514con4d 99 . 2
1615ralrimiva 2789 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  crab 2709  cvv 2956  wsbc 3161   class class class wbr 4212 This theorem is referenced by:  bnj110  29229 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 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213
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