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Metamath Proof Explorer

Theorem exinst01

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 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.)

	Ref	Expression
Hypotheses	exinst01.1	⊢ ∃ 𝑥 𝜓
	exinst01.2	⊢ ( 𝜑 , 𝜓 ▶ 𝜒 )
	exinst01.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	exinst01.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion	exinst01	⊢ ( 𝜑 ▶ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		exinst01.1	⊢ ∃ 𝑥 𝜓
2		exinst01.2	⊢ ( 𝜑 , 𝜓 ▶ 𝜒 )
3		exinst01.3	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4		exinst01.4	⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
5	2	dfvd2i	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
6	1 5 3 4	eexinst01	⊢ ( 𝜑 → 𝜒 )
7	6	dfvd1ir	⊢ ( 𝜑 ▶ 𝜒 )