Theorem exinst11 44599

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
exinst11.1	⊢ (   𝜑   ▶   ∃𝑥𝜓   )
exinst11.2	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
exinst11.3	⊢ (𝜑 → ∀𝑥𝜑)
exinst11.4	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
exinst11	⊢ (   𝜑   ▶   𝜒   )

Proof of Theorem exinst11

Step	Hyp	Ref	Expression
1		exinst11.1	. . . 4 ⊢ (   𝜑   ▶   ∃𝑥𝜓   )
2	1	in1 44544	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
3		exinst11.2	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
4	3	dfvd2i 44558	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5		exinst11.3	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
6		exinst11.4	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
7	2, 4, 5, 6	eexinst11 44500	. 2 ⊢ (𝜑 → 𝜒)
8	7	dfvd1ir 44546	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777  (   wvd1 44542  (   wvd2 44550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-vd1 44543  df-vd2 44551

This theorem is referenced by:  vk15.4jVD  44887