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Theorem exiftru 1657
 Description: A companion rule to ax-gen, valid only if an individual exists. Unlike ax-9 1654, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1546, ax-5 1557 and this theorem alone, not requiring ax-8 1675 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
Hypothesis
Ref Expression
exiftru.1 φ
Assertion
Ref Expression
exiftru xφ

Proof of Theorem exiftru
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 a9ev 1656 . 2 x x = y
2 exiftru.1 . . . 4 φ
32a1i 10 . . 3 (x = yφ)
43eximi 1576 . 2 (x x = yxφ)
51, 4ax-mp 8 1 xφ
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-9 1654 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  19.2  1659
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