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Theorem ax16 1742
 Description: Theorem showing that ax-16 1743 is redundant if ax-17 1465 is included in the axiom system. The important part of the proof is provided by aev 1741. See ax16ALT 1788 for an alternate proof that does not require ax-10 1442 or ax-12 1448. This theorem should not be referenced in any proof. Instead, use ax-16 1743 below so that theorems needing ax-16 1743 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
Assertion
Ref Expression
ax16
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem ax16
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 aev 1741 . 2
2 ax-17 1465 . . . 4
3 sbequ12 1702 . . . . 5
43biimpcd 158 . . . 4
52, 4alimdh 1402 . . 3
62hbsb3 1737 . . . 4
7 stdpc7 1701 . . . 4
86, 2, 7cbv3h 1679 . . 3
95, 8syl6com 35 . 2
101, 9syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1288  wsb 1693 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473 This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694 This theorem is referenced by:  dveeq2  1744  dveeq2or  1745  a16g  1793  exists2  2046
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