Theorem exiftru 1878

Description: Rule of existential generalization, similar to universal generalization ax-gen 1713, but valid only if an individual exists. Its proof requires ax-6 1875 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1713, ax-4 1728 and this theorem alone, not requiring ax-7 1922 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.

Step	Hyp	Ref	Expression
1		ax6ev 1877	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exiftru.1	. . 3 ⊢ 𝜑
3	2	a1i 11	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
4	1, 3	eximii 1754	1 ⊢ ∃𝑥𝜑

Syntax hints:  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  19.2  1879  bj-extru  31843  ac6s6  33150