Theorem allnegex-P5 597

Description: "For all not" is Equivalent to "Does not exist".

Dual of exnegall-P5 598.

Assertion

Ref	Expression
allnegex-P5	⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)

Proof of Theorem allnegex-P5

Step	Hyp	Ref	Expression
1		df-exists-D5.1 596	. . . 4 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	subneg-P3.39.RC 324	. . 3 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ¬ ∀𝑥 ¬ 𝜑)
3		dnegeq-P4.10 418	. . 3 ⊢ (¬ ¬ ∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)
4	2, 3	bitrns-P3.33c.RC 303	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
5	4	bisym-P3.33b.RC 299	1 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃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

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

This theorem is referenced by:  alloverimex-P5  601  qimeqallhalf-P5  609  exiisub-P5  655  lemma-L5.03a  666  gennexw-P6  679  dfnfreealt-P6  683  gennex-P6  738  qcexandr-P6  761