Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  abrexex2 Unicode version

Theorem abrexex2 5796
 Description: Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5779. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1
abrexex2.2
Assertion
Ref Expression
abrexex2
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem abrexex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1609 . . . 4
2 nfcv 2432 . . . . 5
3 nfs1v 2058 . . . . 5
42, 3nfrex 2611 . . . 4
5 sbequ12 1872 . . . . 5
65rexbidv 2577 . . . 4
71, 4, 6cbvab 2414 . . 3
8 df-clab 2283 . . . . 5
98rexbii 2581 . . . 4
109abbii 2408 . . 3
117, 10eqtr4i 2319 . 2
12 df-iun 3923 . . 3
13 abrexex2.1 . . . 4
14 abrexex2.2 . . . 4
1513, 14iunex 5786 . . 3
1612, 15eqeltrri 2367 . 2
1711, 16eqeltri 2366 1
 Colors of variables: wff set class Syntax hints:   wceq 1632  wsb 1638   wcel 1696  cab 2282  wrex 2557  cvv 2801  ciun 3921 This theorem is referenced by:  abexssex  5797  abexex  5798  oprabrexex2  5979  ab2rexex  6016  ab2rexex2  6017 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
 Copyright terms: Public domain W3C validator