Theorem exgen 2006

Description: Rule of existential generalization, similar to universal generalization ax-gen 1827, but valid only if an individual exists. Its proof requires ax-6 1999 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 1827, ax-4 1841 and this theorem alone, not requiring ax-7 2040 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.

Step	Hyp	Ref	Expression
1		idd 25	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜑))
2		exgen.1	. 2 ⊢ 𝜑
3	1, 2	speiv 2004	1 ⊢ ∃𝑥𝜑

This proof depends on syntax axioms:  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-6 1999

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  extru  2007  19.2  2008  vn0OLD  4291  axprg  5394  axnulALT2  35761  axnulg  35870  ac6s6  39198