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Theorem ioran 656
Description: Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 800, anordc 851, or ianordc 792. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ioran (¬ (φ ψ) ↔ (¬ φ ¬ ψ))

Proof of Theorem ioran
StepHypRef Expression
1 pm2.45 644 . . 3 (¬ (φ ψ) → ¬ φ)
2 pm2.46 645 . . 3 (¬ (φ ψ) → ¬ ψ)
31, 2jca 290 . 2 (¬ (φ ψ) → (¬ φ ¬ ψ))
4 simpl 102 . . . . 5 ((¬ φ ¬ ψ) → ¬ φ)
54con2i 545 . . . 4 (φ → ¬ (¬ φ ¬ ψ))
6 simpr 103 . . . . 5 ((¬ φ ¬ ψ) → ¬ ψ)
76con2i 545 . . . 4 (ψ → ¬ (¬ φ ¬ ψ))
85, 7jaoi 623 . . 3 ((φ ψ) → ¬ (¬ φ ¬ ψ))
98con2i 545 . 2 ((¬ φ ¬ ψ) → ¬ (φ ψ))
103, 9impbii 117 1 (¬ (φ ψ) ↔ (¬ φ ¬ ψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wo 616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm4.56  799  dcor  831  3ioran  888  3ori  1181  unssdif  3149  difundi  3166  sotricim  4034  sotritrieq  4036  en2lp  4216  poxp  5775  nntri2  5988  letr  6699  nnexmid  7004
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