Theorem gennfrcl-L6 812

Description: Closed Form of gennfr-P6 734.

Assertion

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

Proof of Theorem gennfrcl-L6

Step	Hyp	Ref	Expression
1		alloverimex-P5.CL 604	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑))
2		exgenall-P6 732	. . . 4 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	syl-P3.24 259	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
5		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6	5	bisym-P3.33b.RC 299	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ Ⅎ𝑥𝜑)
7	4, 6	subimr2-P4.RC 543	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

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-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:  exgennfrcl-L6  814  nfrall2d-P6  819