Theorem exinst 45075

Description: Existential Instantiation. Virtual deduction form of exlimexi 44975. (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 45036	. 2 ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))
4	1, 3	exlimexi 44975	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786  (   wvd2 45028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-vd2 45029

This theorem is referenced by:  sb5ALTVD  45363