Theorem imnan 690

Description: Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)

Assertion

Ref	Expression
imnan	⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))

Proof of Theorem imnan

Step	Hyp	Ref	Expression
1		pm3.2im 637	. . . 4 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
2	1	imp 124	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓))
3	2	con2i 627	. 2 ⊢ ((𝜑 → ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
4		pm3.2 139	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
5	4	con3rr3 633	. 2 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜑 → ¬ 𝜓))
6	3, 5	impbii 126	1 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  imnani  691  nan  692  pm3.24  693  imanst  888  ianordc  899  pm5.17dc  904  dn1dc  960  xorbin  1384  xordc1  1393  alinexa  1603  dfrex2dc  2468  ralinexa  2504  rabeq0  3452  disj  3471  minel  3484  disjsn  3654  sotricim  4323  poirr2  5021  funun  5260  imadiflem  5295  imadif  5296  brprcneu  5508  2omotaplemap  7255  prltlu  7485  caucvgprlemnbj  7665  caucvgprprlemnbj  7691  suplocexprlemmu  7716  xrltnsym2  9793  fzp1nel  10103  fsumsplit  11414  sumsplitdc  11439  phiprmpw  12221  odzdvds  12244  pcdvdsb  12318  lgsne0  14409  bj-nnor  14456