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Theorem anandi 562
Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
Assertion
Ref Expression
anandi ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem anandi
StepHypRef Expression
1 anidm 391 . . 3 ((𝜑𝜑) ↔ 𝜑)
21anbi1i 451 . 2 (((𝜑𝜑) ∧ (𝜓𝜒)) ↔ (𝜑 ∧ (𝜓𝜒)))
3 an4 558 . 2 (((𝜑𝜑) ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
42, 3bitr3i 185 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anandi3  958  moanim  2049  difundi  3296  inrab  3316  uniin  3724  xpcom  5053  fin  5277  fndmin  5493  nnaord  6371  ixpin  6583  ltexprlemdisj  7378  bldisj  12476  blininf  12499
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