Theorem andasim-P3.46a 356

Description: '∧' in Terms of '→'.

This is the re-derived Chapter 2 definition. Only andasimint-P3.46b 357 is deducible with intuitionist logic.

Assertion

Ref	Expression
andasim-P3.46a	⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))

Proof of Theorem andasim-P3.46a

Step	Hyp	Ref	Expression
1		andasim-P3.46-L1 354	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓))
2		andasim-P3.46-L2 355	. 2 ⊢ (¬ (𝜑 → ¬ 𝜓) → (𝜑 ∧ 𝜓))
3	1, 2	rcp-NDBII0 239	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132

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

This theorem is referenced by:  nfrand-P6  690  qcexandr-P6  761  qcexandl-P6  762  psubneg-P6-L1  787  psuband-P6  792  nfrandd-P6  816