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Theorem equidqe 2209
Description: equid 1683 with existential quantifier without using ax-4 2171 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)
Assertion
Ref Expression
equidqe  |-  -.  A. y  -.  x  =  x

Proof of Theorem equidqe
StepHypRef Expression
1 ax9from9o 2184 . 2  |-  -.  A. y  -.  y  =  x
2 ax-8 1682 . . . . 5  |-  ( y  =  x  ->  (
y  =  x  ->  x  =  x )
)
32pm2.43i 45 . . . 4  |-  ( y  =  x  ->  x  =  x )
43con3i 129 . . 3  |-  ( -.  x  =  x  ->  -.  y  =  x
)
54alimi 1565 . 2  |-  ( A. y  -.  x  =  x  ->  A. y  -.  y  =  x )
61, 5mto 169 1  |-  -.  A. y  -.  x  =  x
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1546
This theorem is referenced by:  ax4sp1  2210  equidq  2211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-8 1682  ax-6o 2173  ax-9o 2174
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