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.

Step	Hyp	Ref	Expression
1		a9ev 1656	. 2 ⊢ ∃x x = y
2		exiftru.1	. . . 4 ⊢ φ
3	2	a1i 10	. . 3 ⊢ (x = y → φ)
4	3	eximi 1576	. 2 ⊢ (∃x x = y → ∃xφ)
5	1, 4	ax-mp 5	1 ⊢ ∃xφ

Syntax hints:  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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