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Theorem axrep1 4564
Description: The version of the Axiom of Replacement used in the Metamath Solitaire applet http://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 4563 -> axrep1 4564 -> axrep2 4565 -> axrepnd 8990 -> zfcndrep 9013 = ax-rep 4563. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
axrep1
Distinct variable groups:   ,   , ,

Proof of Theorem axrep1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elequ2 1823 . . . . . . . . 9
21anbi1d 704 . . . . . . . 8
32exbidv 1714 . . . . . . 7
43bibi2d 318 . . . . . 6
54albidv 1713 . . . . 5
65exbidv 1714 . . . 4
76imbi2d 316 . . 3
8 ax-rep 4563 . . . 4
9 19.3v 1755 . . . . . . . 8
109imbi1i 325 . . . . . . 7
1110albii 1640 . . . . . 6
1211exbii 1667 . . . . 5
1312albii 1640 . . . 4
14 nfv 1707 . . . . . . 7
15 nfe1 1840 . . . . . . 7
1614, 15nfbi 1934 . . . . . 6
1716nfal 1947 . . . . 5
18 nfv 1707 . . . . 5
19 elequ2 1823 . . . . . . 7
209anbi2i 694 . . . . . . . . 9
2120exbii 1667 . . . . . . . 8
2221a1i 11 . . . . . . 7
2319, 22bibi12d 321 . . . . . 6
2423albidv 1713 . . . . 5
2517, 18, 24cbvex 2022 . . . 4
268, 13, 253imtr3i 265 . . 3
277, 26chvarv 2014 . 2
282719.35ri 1690 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:  axrep2  4565
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
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