Theorem allic-P6 745

Description: Introduction of Universal Quantifier as Consequent (non-freeness condition).

This proposition is equivalent to the '∀' introduction rule in the natural deduction system.

Hypotheses

Ref	Expression
allic-P6.1	⊢ Ⅎ𝑥𝜑
allic-P6.2	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
allic-P6	⊢ (𝜑 → ∀𝑥𝜓)

Proof of Theorem allic-P6

Step	Hyp	Ref	Expression
1		allic-P6.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfrgen-P6 733	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		allic-P6.2	. . 3 ⊢ (𝜑 → 𝜓)
4	3	alloverim-P5.RC.GEN 592	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
5	2, 4	syl-P3.24.RC 260	1 ⊢ (𝜑 → ∀𝑥𝜓)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  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-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

This theorem is referenced by:  alloverim-P5.GENF  747  alloverimex-P5.GENF  748  cbvall-P6-L1  750  nfrall2d-P6  819  nfrex2d-P6  820  ndalli-P6  822