Theorem exnegall-P7 1046

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

This statement is not deducible with intuitionist logic.

Assertion

Ref	Expression
exnegall-P7	⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)

Proof of Theorem exnegall-P7

Step	Hyp	Ref	Expression
1		dfexists-P7 959	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ¬ 𝜑)
2		dnegeq-P4.10 418	. . . 4 ⊢ (¬ ¬ 𝜑 ↔ 𝜑)
3	2	suball-P7.RC 974	. . 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  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-false-D2.5 158  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  allasex-P7  1048  qimeqex-P7-L1  1054  nfrnegconv-P8  1110