Theorem exinst 45374

Description: Existential Instantiation. Virtual deduction form of exlimexi 45274. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
exinst.1	⊢ (𝜓 → ∀𝑥𝜓)
exinst.2	⊢ (   ∃𝑥𝜑   ,   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
exinst	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem exinst

Step	Hyp	Ref	Expression
1		exinst.1	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2		exinst.2	. . 3 ⊢ (   ∃𝑥𝜑   ,   𝜑   ▶   𝜓   )
3	2	dfvd2i 45335	. 2 ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))
4	1, 3	exlimexi 45274	1 ⊢ (∃𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  (   wvd2 45327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-vd2 45328

This theorem is used by:  sb5ALTVD  45662