Theorem allic-P7 1007

Description: Introduce Universal Quantifier as Consequent. †

The inference form is alli-P7r 990.

Hypotheses

Ref	Expression
allic-P7.1	⊢ Ⅎ𝑥𝛾
allic-P7.2	⊢ Ⅎ𝑥𝜑
allic-P7.3	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
allic-P7	⊢ (𝛾 → (𝜑 → ∀𝑥𝜓))

Proof of Theorem allic-P7

Step	Hyp	Ref	Expression
1		allic-P7.1	. . . 4 ⊢ Ⅎ𝑥𝛾
2		allic-P7.2	. . . 4 ⊢ Ⅎ𝑥𝜑
3	1, 2	ndnfrand-P7.4.RC 877	. . 3 ⊢ Ⅎ𝑥(𝛾 ∧ 𝜑)
4		allic-P7.3	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
5	4	import-P3.34a.RC 306	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜓)
6	3, 5	alli-P7 947	. 2 ⊢ ((𝛾 ∧ 𝜑) → ∀𝑥𝜓)
7	6	rcp-NDIMI2 224	1 ⊢ (𝛾 → (𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  allic-P7.VR1of2  1008  allic-P7.VR2of2  1009  allic-P7.VR12of2  1010