Theorem dfnfreeint-P7 969

Description: Necessary Condition for (i.e. "If" part of) df-nfree-D6.1 682 Derived From Natural Deduction Rules. †

Only this direction is deducible with intuitionist logic.

Assertion

Ref	Expression
dfnfreeint-P7	⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem dfnfreeint-P7

Step	Hyp	Ref	Expression
1		orcomm-P3.37 319	. . . 4 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑))
2		allnegex-P7 958	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	suborl-P3.43a.RC 347	. . . 4 ⊢ ((∀𝑥 ¬ 𝜑 ∨ ∀𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
4	1, 3	bitrns-P3.33c.RC 303	. . 3 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
5	4	rcp-NDBIEF0 240	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑))
6		imasorint-P4.32b 488	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
7		dfnfreealtif-P7 964	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
8	5, 6, 7	dsyl-P3.25.RC 262	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ∨ 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)