Theorem allicv-P5 614

Description: Introduction of Universal Quantifier as Consequent. (variable restriction).

'𝑥' may occur in '𝜓' but not '𝜑'.

This is a weak version of the '∀' intruduction rule in the natural deduction system. The version with a non-freeness condition is allic-P6 745.

Hypothesis

Ref	Expression
allicv-P5.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
allicv-P5	⊢ (𝜑 → ∀𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Proof of Theorem allicv-P5

Step	Hyp	Ref	Expression
1		ax-L5 17	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		allicv-P5.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alloverim-P5.RC.GEN 592	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
4	1, 3	syl-P3.24.RC 260	1 ⊢ (𝜑 → ∀𝑥𝜓)

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

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

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  alloverim-P5.GENV  621  alloverimex-P5.GENV  622  specpsub-P6  721