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Theorem equidqe 2173
Description: equid 1676 with existential quantifier without using ax-4 2135 or ax-17 1616. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)
Assertion
Ref Expression
equidqe ¬ y ¬ x = x

Proof of Theorem equidqe
StepHypRef Expression
1 ax9from9o 2148 . 2 ¬ y ¬ y = x
2 ax-8 1675 . . . . 5 (y = x → (y = xx = x))
32pm2.43i 43 . . . 4 (y = xx = x)
43con3i 127 . . 3 x = x → ¬ y = x)
54alimi 1559 . 2 (y ¬ x = xy ¬ y = x)
61, 5mto 167 1 ¬ y ¬ x = x
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-8 1675  ax-6o 2137  ax-9o 2138
This theorem is referenced by:  ax4sp1  2174  equidq  2175
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