Theorem idempotall-P8 1093

Description: Idempotency Law for '∀𝑥'. †

Assertion

Ref	Expression
idempotall-P8	⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem idempotall-P8

Step	Hyp	Ref	Expression
1		alle-P7.CL 942	. 2 ⊢ (∀𝑥∀𝑥𝜑 → ∀𝑥𝜑)
2		ndnfrall1-P7.7 832	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	nfrgen-P7.CL 930	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
4	1, 3	rcp-NDBII0 239	1 ⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8   ↔ wff-bi 104

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: (None)