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Theorem ax16b 4218
Description: This theorem shows that axiom ax-16 2096 is redundant in the presence of theorem dtru 4217, which states simply that at least two things exist. This justifies the remark at http://us.metamath.org/mpeuni/mmzfcnd.html#twoness (which links to this theorem). (Contributed by NM, 7-Nov-2006.)
Assertion
Ref Expression
ax16b  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax16b
StepHypRef Expression
1 dtru 4217 . 2  |-  -.  A. x  x  =  y
21pm2.21i 123 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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