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Definition df-an 396
Description: Define conjunction (logical "and"). Definition of [Margaris] p. 49. When both the left and right operand are true, the result is true; when either is false, the result is false. For example, it is true that (2 = 2 ∧ 3 = 3). After we define the constant true (df-tru 1536) and the constant false (df-fal 1546), we will be able to prove these truth table values: ((⊤ ∧ ⊤) ↔ ⊤) (truantru 1566), ((⊤ ∧ ⊥) ↔ ⊥) (truanfal 1567), ((⊥ ∧ ⊤) ↔ ⊥) (falantru 1568), and ((⊥ ∧ ⊥) ↔ ⊥) (falanfal 1569).

This is our first use of the biconditional connective in a definition; we use the biconditional connective in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute ¬ (𝜑 → ¬ 𝜓) for (𝜑𝜓), we end up with an instance of previously proved theorem biid 261. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-or 845), (wi 4), (df-nan 1485), and (df-xor 1505). (Contributed by NM, 5-Jan-1993.)

Assertion
Ref Expression
df-an ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))

Detailed syntax breakdown of Definition df-an
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 wps . . 3 wff 𝜓
31, 2wa 395 . 2 wff (𝜑𝜓)
42wn 3 . . . 4 wff ¬ 𝜓
51, 4wi 4 . . 3 wff (𝜑 → ¬ 𝜓)
65wn 3 . 2 wff ¬ (𝜑 → ¬ 𝜓)
73, 6wb 205 1 wff ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
Colors of variables: wff setvar class
This definition is referenced by:  pm4.63  397  imnan  399  imp  406  ex  412  dfbi2  474  pm5.32  573  pm4.54  983  nfand  1892  nfan1  2185  dfac5lem4  10118  kmlem3  10144  nolt02o  27569  axrepprim  35195  axunprim  35196  axregprim  35198  axinfprim  35199  axacprim  35200  orddif0suc  42568  dfxor4  43067  df3an2  43070  expandan  43597  ismnuprim  43603  pm11.52  43696
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