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Theorem an12 561
Description: Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
Assertion
Ref Expression
an12 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜓 ∧ (𝜑𝜒)))

Proof of Theorem an12
StepHypRef Expression
1 ancom 266 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
21anbi1i 458 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜓𝜑) ∧ 𝜒))
3 anass 401 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
4 anass 401 . 2 (((𝜓𝜑) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))
52, 3, 43bitr3i 210 1 ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜓 ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  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
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  an32  562  an13  563  an12s  565  an4  586  ceqsrexv  2867  rmoan  2937  2reuswapdc  2941  reuind  2942  2rmorex  2943  sbccomlem  3037  elunirab  3822  rexxfrd  4463  opeliunxp  4681  elres  4943  resoprab  5970  ov6g  6011  opabex3d  6121  opabex3  6122  xpassen  6829  distrnqg  7385  distrnq0  7457  rexuz2  9579  2clim  11304  issubrg  13342  isbasis2g  13476  tgval2  13482
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