Theorem nfrnfr-P6 821

Description: ENF Over ENF Predicate.

Interpreted semantically, this says that the effective non-freeness predicate does not depend on the value assigned to the indicated variable, '𝑥'. This means effective non-freeness is a property of a WFF that will either be true for every '𝑥' assignment or false for every '𝑥' assignment. We expect this since ENF is a property that we should be able to determine grammatically.

Assertion

Ref	Expression
nfrnfr-P6	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfrnfr-P6

Step	Hyp	Ref	Expression
1		nfrex1-P6 742	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2		nfrall1-P6 741	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	nfrim-P6 689	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
4		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	4	nfrleq-P6 687	. 2 ⊢ (Ⅎ𝑥Ⅎ𝑥𝜑 ↔ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑))
6	3, 5	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃wff-exists 595  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-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:  ndnfrnfr-P7.12  837