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Theorem axnulALT 4579
Description: Alternate proof of axnul 4580, proved directly from ax-rep 4563 using none of the equality axioms ax-7 1790 through ax-c14 2222 provided we accept sp 1859 as an axiom. Replace sp 1859 with the obsolete ax-c5 2214 to see this in 'show traceback'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axnulALT
Distinct variable group:   ,

Proof of Theorem axnulALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-rep 4563 . . 3
2 sp 1859 . . . . . 6
32con2i 120 . . . . 5
4 df-ex 1613 . . . . 5
53, 4sylibr 212 . . . 4
6 fal 1402 . . . . . 6
7 sp 1859 . . . . . 6
86, 7mto 176 . . . . 5
98pm2.21i 131 . . . 4
105, 9mpg 1620 . . 3
111, 10mpg 1620 . 2
128intnan 914 . . . . . 6
1312nex 1627 . . . . 5
1413nbn 347 . . . 4
1514albii 1640 . . 3
1615exbii 1667 . 2
1711, 16mpbir 209 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395   wfal 1400  E.wex 1612  e.wcel 1818
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-12 1854  ax-rep 4563
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-fal 1401  df-ex 1613
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