Theorem exinst01 41331

Description: Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E ∃ in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
exinst01.1	⊢ ∃𝑥𝜓
exinst01.2	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
exinst01.3	⊢ (𝜑 → ∀𝑥𝜑)
exinst01.4	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
exinst01	⊢ (   𝜑   ▶   𝜒   )

Proof of Theorem exinst01

Step	Hyp	Ref	Expression
1		exinst01.1	. . 3 ⊢ ∃𝑥𝜓
2		exinst01.2	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
3	2	dfvd2i 41291	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4		exinst01.3	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		exinst01.4	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
6	1, 3, 4, 5	eexinst01 41232	. 2 ⊢ (𝜑 → 𝜒)
7	6	dfvd1ir 41279	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  (   wvd1 41275  (   wvd2 41283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-vd1 41276  df-vd2 41284

This theorem is referenced by:  vk15.4jVD  41620