Theorem allasex-P5 599

Description: Universal Quantification in Terms of Existential Quantification.

Dual of df-exists-D5.1 596.

Assertion

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

Proof of Theorem allasex-P5

Step	Hyp	Ref	Expression
1		exnegall-P5 598	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2	1	subneg-P3.39.RC 324	. . 3 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥𝜑)
3		dnegeq-P4.10 418	. . 3 ⊢ (¬ ¬ ∀𝑥𝜑 ↔ ∀𝑥𝜑)
4	2, 3	bitrns-P3.33c.RC 303	. 2 ⊢ (¬ ∃𝑥 ¬ 𝜑 ↔ ∀𝑥𝜑)
5	4	bisym-P3.33b.RC 299	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

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

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:  spec-P6  719  psubneg-P6-L1  787