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Theorem axrep2 4134
 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:   ,,
Allowed substitution hints:   (,,)

Proof of Theorem axrep2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfe1 1707 . . . . 5
2 nfv 1605 . . . . 5
31, 2nfim 1771 . . . 4
43nfex 1769 . . 3
5 elequ2 1690 . . . . . . . . 9
65anbi1d 685 . . . . . . . 8
76exbidv 1612 . . . . . . 7
87bibi2d 309 . . . . . 6
98albidv 1611 . . . . 5
109imbi2d 307 . . . 4
1110exbidv 1612 . . 3
12 axrep1 4133 . . 3
134, 11, 12chvar 1929 . 2
14 sp 1717 . . . . . . . 8
1514imim1i 54 . . . . . . 7
1615alimi 1546 . . . . . 6
1716eximi 1563 . . . . 5
18 nfv 1605 . . . . . 6
19 nfa1 1758 . . . . . . . 8
20 nfv 1605 . . . . . . . 8
2119, 20nfim 1771 . . . . . . 7
2221nfal 1768 . . . . . 6
23 equequ2 1650 . . . . . . . 8
2423imbi2d 307 . . . . . . 7
2524albidv 1611 . . . . . 6
2618, 22, 25cbvex 1928 . . . . 5
2717, 26sylib 188 . . . 4
2827imim1i 54 . . 3
2928eximi 1563 . 2
3013, 29ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1527  wex 1528   wceq 1623   wcel 1685 This theorem is referenced by:  axrep3  4135  axrepndlem1  8210 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-cleq 2277  df-clel 2280
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