Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrep2 Unicode version

Theorem axrep2 4565
 Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on . (Contributed by NM, 15-Aug-2003.)
Assertion
Ref Expression
axrep2
Distinct variable group:   ,,

Proof of Theorem axrep2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfe1 1840 . . . . 5
2 nfv 1707 . . . . 5
31, 2nfim 1920 . . . 4
43nfex 1948 . . 3
5 elequ2 1823 . . . . . . . . 9
65anbi1d 704 . . . . . . . 8
76exbidv 1714 . . . . . . 7
87bibi2d 318 . . . . . 6
98albidv 1713 . . . . 5
109imbi2d 316 . . . 4
1110exbidv 1714 . . 3
12 axrep1 4564 . . 3
134, 11, 12chvar 2013 . 2
14 sp 1859 . . . . . . 7
1514imim1i 58 . . . . . 6
1615alimi 1633 . . . . 5
1716eximi 1656 . . . 4
18 nfv 1707 . . . . 5
19 nfa1 1897 . . . . . . 7
20 nfv 1707 . . . . . . 7
2119, 20nfim 1920 . . . . . 6
2221nfal 1947 . . . . 5
23 equequ2 1799 . . . . . . 7
2423imbi2d 316 . . . . . 6
2524albidv 1713 . . . . 5
2618, 22, 25cbvex 2022 . . . 4
2717, 26sylib 196 . . 3
2827imim1i 58 . 2
2913, 28eximii 1658 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612 This theorem is referenced by:  axrep3  4566  axrepndlem1  8988 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-rep 4563 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617
 Copyright terms: Public domain W3C validator