Theorem nfrexall-P8.CL 1090

Description: Closed Form of nfrexall-P8 1086. †

Hypothesis

Ref	Expression
nfrexall-P8.CL.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexall-P8.CL	⊢ (∃𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem nfrexall-P8.CL

Step	Hyp	Ref	Expression
1		nfrexall-P8.CL.1	. 2 ⊢ Ⅎ𝑥𝜑
2		rcp-NDASM1of1 192	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥𝜑)
3	1, 2	nfrexall-P8 1086	1 ⊢ (∃𝑥𝜑 → ∀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-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:  nfrexall-P8.CL.VR  1091  qswap-P8  1105