Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax16 Unicode version

Theorem ax16 1925
 Description: Theorem showing that ax-16 1926 is redundant if ax-17 1628 is included in the axiom system. The important part of the proof is provided by aev 1923. See ax16ALT 1995 for an alternate proof that does not require ax-10 1678 or ax-12o 1664. This theorem should not be referenced in any proof. Instead, use ax-16 1926 below so that theorems needing ax-16 1926 can be more easily identified. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ax16
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem ax16
StepHypRef Expression
1 aev 1923 . 2
2 nfv 1629 . . . 4
3 sbequ12 1892 . . . . 5
43biimpcd 217 . . . 4
52, 4alimd 1705 . . 3
62nfs1 1921 . . . 4
7 stdpc7 1891 . . . 4
86, 2, 7cbv3 1874 . . 3
95, 8syl6com 33 . 2
101, 9syl 17 1
 Colors of variables: wff set class Syntax hints:   wi 6  wal 1532   wceq 1619  wsb 1882 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883
 Copyright terms: Public domain W3C validator