Theorem nfrgencl-L6 811

Description: Closed Form of nfrgen-P6 733.

Assertion

Ref	Expression
nfrgencl-L6	⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nfrgencl-L6

Step	Hyp	Ref	Expression
1		exi-P6 718	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∃𝑥𝜑))
3		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	rcp-NDBIEF0 240	. 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
5	2, 4	syl-P3.24 259	1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

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-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:  nfrexgencl-L6  813  nfrall2d-P6  819