Theorem gennfrcl-P7 963

Description: General for ⇒ ENF in (closed form). †

Assertion

Ref	Expression
gennfrcl-P7	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem gennfrcl-P7

Step	Hyp	Ref	Expression
1		ndnfrall1-P7.7 832	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥∀𝑥𝜑)
3		ndnfrall1-P7.7 832	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → ∀𝑥𝜑)
4		alle-P7.CL 942	. . . . 5 ⊢ (∀𝑥𝜑 → 𝜑)
5	4	rcp-NDIMP0addall 207	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → 𝜑))
6		alle-P7.CL 942	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
7	5, 6	ndbii-P3.13 178	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 ↔ 𝜑))
8	3, 7	ndnfrleq-P7.11 836	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (Ⅎ𝑥∀𝑥𝜑 ↔ Ⅎ𝑥𝜑))
9	2, 8	bimpf-P4 531	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  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-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  dfnfreealtif-P7  964  nfrcond-P8  1108