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Theorem andir 732
Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
andir (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Proof of Theorem andir
StepHypRef Expression
1 andi 731 . 2 ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
2 ancom 253 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
3 ancom 253 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 ancom 253 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4orbi12i 681 . 2 (((𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
61, 2, 53bitr4i 201 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wo 629
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-io 630
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  anddi  734  dcan  842  excxor  1269  xordc1  1284  sbequilem  1719  rexun  3123  rabun2  3216  reuun2  3220  xpundir  4397  coundi  4822  mptun  5029  tpostpos  5879  ltxr  8695
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