Theorem exinst 45701

Description: Existential Instantiation. Virtual deduction form of exlimexi 45601. (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 45662	. 2 ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))
4	1, 3	exlimexi 45601	1 ⊢ (∃𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  (   wvd2 45654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-vd2 45655

This theorem is used by:  sb5ALTVD  45989