Theorem exnegall-P5 598

Description: "Exists a negative" is Equivalent to "Not for all".

Dual of allnegex-P5 597.

Assertion

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

Proof of Theorem exnegall-P5

Step	Hyp	Ref	Expression
1		df-exists-D5.1 596	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ¬ 𝜑)
2		dnegeq-P4.10 418	. . . 4 ⊢ (¬ ¬ 𝜑 ↔ 𝜑)
3	2	suballinf-P5 594	. . 3 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ∀𝑥𝜑)
4	3	subneg-P3.39.RC 324	. 2 ⊢ (¬ ∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5	1, 4	bitrns-P3.33c.RC 303	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  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:  allasex-P5  599  qimeqex-P5-L1  610  gennallw-P6  678  qcexandr-P6  761  qcexandl-P6  762