Theorem nfrexgencl-L6 813

Description: Closed Form of nfrexgen-P6 735.

Assertion

Ref	Expression
nfrexgencl-L6	⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

Proof of Theorem nfrexgencl-L6

Step	Hyp	Ref	Expression
1		nfrgencl-L6 811	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2	1	trnsp-P3.31b 282	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 → (¬ ∀𝑥 ¬ 𝜑 → 𝜑))
3		nfrneg-P6 688	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
4		df-exists-D5.1 596	. . . 4 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
5	4	bisym-P3.33b.RC 299	. . 3 ⊢ (¬ ∀𝑥 ¬ 𝜑 ↔ ∃𝑥𝜑)
6	5	subiml-P3.40a.RC 326	. 2 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))
7	2, 3, 6	subimd2-P4.RC 545	1 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → 𝜑))

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-L5 17  ax-L6 18  ax-L7 19  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:  nfrex2d-P6  820