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Theorem andir 731
Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
andir (((φ ψ) χ) ↔ ((φ χ) (ψ χ)))

Proof of Theorem andir
StepHypRef Expression
1 andi 730 . 2 ((χ (φ ψ)) ↔ ((χ φ) (χ ψ)))
2 ancom 253 . 2 (((φ ψ) χ) ↔ (χ (φ ψ)))
3 ancom 253 . . 3 ((φ χ) ↔ (χ φ))
4 ancom 253 . . 3 ((ψ χ) ↔ (χ ψ))
53, 4orbi12i 680 . 2 (((φ χ) (ψ χ)) ↔ ((χ φ) (χ ψ)))
61, 2, 53bitr4i 201 1 (((φ ψ) χ) ↔ ((φ χ) (ψ χ)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 628
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 629
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  anddi  733  dcan  841  excxor  1268  xordc1  1281  sbequilem  1716  rexun  3117  rabun2  3210  reuun2  3214  xpundir  4340  coundi  4765  mptun  4972  tpostpos  5820  ltxr  8465
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