Theorem exgennfrcl-L6 814

Description: Closed Form of exgennfr-P6 736.

Assertion

Ref	Expression
exgennfrcl-L6	⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem exgennfrcl-L6

Step	Hyp	Ref	Expression
1		lemma-L5.01a 600	. . . . 5 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2	1	rcp-NDBIEF0 240	. . . 4 ⊢ ((∃𝑥𝜑 → 𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	2	alloverim-P5.RC.GEN 592	. . 3 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4		gennfrcl-L6 812	. . 3 ⊢ (∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → Ⅎ𝑥 ¬ 𝜑)
5	3, 4	syl-P3.24.RC 260	. 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) → Ⅎ𝑥 ¬ 𝜑)
6		nfrneg-P6 688	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
7	5, 6	subimr2-P4.RC 543	1 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → 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-L10 27

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:  nfrex2d-P6  820