Theorem dfnfree-P7 968

Description: df-nfree-D6.1 682 Derived From Natural Deduction Rules.

This definition is not valid with intuitionist logic.

Assertion

Ref	Expression
dfnfree-P7	⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Proof of Theorem dfnfree-P7

Step	Hyp	Ref	Expression
1		dfnfreealt-P7 967	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		imasor-P4.32a 487	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
3		allnegex-P7 958	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
4	3	bisym-P3.33b.RC 299	. . 3 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
5	4	suborl-P3.43a.RC 347	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
6		orcomm-P3.37 319	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑) ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
7	1, 2, 5, 6	tbitrns-P4.17.RC 431	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∨ wff-or 144  ∃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-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-false-D2.5 158  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by: (None)