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Theorem exgen 1972
Description: Rule of existential generalization, similar to universal generalization ax-gen 1792, but valid only if an individual exists. Its proof requires ax-6 1965 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen 1792, ax-4 1806 and this theorem alone, not requiring ax-7 2005 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
Hypothesis
Ref Expression
exgen.1 𝜑
Assertion
Ref Expression
exgen 𝑥𝜑

Proof of Theorem exgen
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 idd 24 . 2 (𝑥 = 𝑦 → (𝜑𝜑))
2 exgen.1 . 2 𝜑
31, 2speiv 1970 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-6 1965
This theorem depends on definitions:  df-bi 207  df-ex 1777
This theorem is referenced by:  extru  1973  19.2  1974  vn0  4351  axnulg  35085  ac6s6  38159
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