Theorem allnegex-P7 958

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

This statement is deducible with intuitionist logic. It's dual, given by exnegall-P7 1046, is not.

Assertion

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

Proof of Theorem allnegex-P7

Step	Hyp	Ref	Expression
1		allnegex-P7-L1 956	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
2		allnegex-P7-L2 957	. 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ 𝜑)
3	1, 2	rcp-NDBII0 239	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:  dfexists-P7  959  dfexistsint-P7  960  dfnfree-P7  968  dfnfreeint-P7  969  allnegex-P7r  1045  nfrnegconv-P8  1110