Theorem genall-P6 737

Description: The WFF '∀𝑥𝜑' is General For '𝑥'.

See genallw-P6 676 for a version that requires only FOL axioms.

Assertion

Ref	Expression
genall-P6	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Proof of Theorem genall-P6

Step	Hyp	Ref	Expression
1		gennall-P6 730	. . . 4 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
2	1	gennfr-P6 734	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥𝜑
3		nfrneg-P6 688	. . 3 ⊢ (Ⅎ𝑥 ¬ ∀𝑥𝜑 ↔ Ⅎ𝑥∀𝑥𝜑)
4	2, 3	bimpf-P4.RC 532	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
5	4	nfrgen-P6 733	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → 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-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:  gennex-P6  738  nfrall1-P6  741