Theorem idempotallnex-P8 1099

Description: Idempotency Law for '∀𝑥' over '¬ ∃𝑥'. †

Assertion

Ref	Expression
idempotallnex-P8	⊢ (∀𝑥 ¬ ∃𝑥𝜑 ↔ ¬ ∃𝑥𝜑)

Proof of Theorem idempotallnex-P8

Step	Hyp	Ref	Expression
1		alle-P7.CL 942	. 2 ⊢ (∀𝑥 ¬ ∃𝑥𝜑 → ¬ ∃𝑥𝜑)
2		ndnfrex1-P7.8 833	. . . 4 ⊢ Ⅎ𝑥∃𝑥𝜑
3	2	ndnfrneg-P7.2.RC 875	. . 3 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑
4	3	nfrgen-P7.CL 930	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)
5	1, 4	rcp-NDBII0 239	1 ⊢ (∀𝑥 ¬ ∃𝑥𝜑 ↔ ¬ ∃𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   ↔ wff-bi 104  ∃wff-exists 595

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)