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Theorem exiftru 1713
Description: A companion rule to ax-gen, valid only if an individual exists. Unlike ax-6 1710, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1592, ax-4 1603 and this theorem alone, not requiring ax-7 1730 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  |-  ph
Assertion
Ref Expression
exiftru  |-  E. x ph

Proof of Theorem exiftru
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1712 . 2  |-  E. x  x  =  y
2 exiftru.1 . . 3  |-  ph
32a1i 11 . 2  |-  ( x  =  y  ->  ph )
41, 3eximii 1628 1  |-  E. x ph
Colors of variables: wff setvar class
Syntax hints:   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-6 1710
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by:  19.2  1714  ac6s6  29133
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