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Theorem axc16b 4595
Description: This theorem shows that axiom ax-c16 32464 is redundant in the presence of theorem dtru 4594, 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). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
Assertion
Ref Expression
axc16b  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem axc16b
StepHypRef Expression
1 dtru 4594 . 2  |-  -.  A. x  x  =  y
21pm2.21i 135 1  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-nul 4534  ax-pow 4581
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
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