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List of Syntax, Axioms (ax-) and Definitions (df-)
RefExpression (see link for any distinct variable requirements)
wn 3wff ¬ φ
wi 4wff (φψ)
ax-1 5(φ → (ψφ))
ax-2 6((φ → (ψχ)) → ((φψ) → (φχ)))
ax-mp 7φ    &   (φψ)       ψ
wa 95wff (φ ψ)
wb 96wff (φψ)
ax-ia1 97((φ ψ) → φ)
ax-ia2 98((φ ψ) → ψ)
ax-ia3 99(φ → (ψ → (φ ψ)))
df-bi 108(((φψ) → ((φψ) (ψφ))) (((φψ) (ψφ)) → (φψ)))
ax-in1 527((φ → ¬ φ) → ¬ φ)
ax-in2 528φ → (φψ))
wo 608wff (φ ψ)
ax-io 609(((φ χ) → ψ) ↔ ((φψ) (χψ)))
wdc 717wff DECID φ
df-dc 718(DECID φ ↔ (φ ¬ φ))
w3o 842wff (φ ψ χ)
w3a 843wff (φ ψ χ)
df-3or 844((φ ψ χ) ↔ ((φ ψ) χ))
df-3an 845((φ ψ χ) ↔ ((φ ψ) χ))
wtru 1188wff
wfal 1189wff
df-tru 1191( ⊤ ↔ (φφ))
df-fal 1192( ⊥ ↔ ¬ ⊤ )
wxo 1207wff (φψ)
df-xor 1208((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
wal 1267wff xφ
ax-5 1268(x(φψ) → (xφxψ))
ax-7 1269(xyφyxφ)
ax-gen 1270φ       xφ
wnf 1281wff xφ
df-nf 1282(Ⅎxφx(φxφ))
wex 1314wff xφ
ax-ie1 1315(xφxxφ)
ax-ie2 1316(x(ψxψ) → (x(φψ) ↔ (xφψ)))
cv 1324class x
wceq 1325wff A = B
wcel 1327wff A B
ax-8 1329(x = y → (x = zy = z))
ax-10 1330(x x = yy y = x)
ax-11 1331(x = y → (yφx(x = yφ)))
ax-i12 1332(z z = x (z z = y z(x = yz x = y)))
ax-bnd 1333(z z = x (z z = y xz(x = yz x = y)))
ax-4 1334(xφφ)
ax-13 1338(x = y → (x zy z))
ax-14 1339(x = y → (z xz y))
ax-17 1352(φxφ)
ax-i9 1356x x = y
ax-ial 1361(xφxxφ)
ax-i5r 1362((xφxψ) → x(xφψ))
ax-10o 1517(x x = y → (xφyφ))
wsbc 1555wff [A / x]φ
df-sb 1557([y / x]φ ↔ ((x = yφ) x(x = y φ)))
ax-16 1605(x x = y → (φxφ))
ax-11o 1614x x = y → (x = y → (φx(x = yφ))))
weu 1804wff ∃!xφ
wmo 1805wff ∃*xφ
df-eu 1807(∃!xφyx(φx = y))
df-mo 1808(∃*xφ ↔ (xφ∃!xφ))
ax-ext 1850(z(z xz y) → x = y)
cab 1853class {xφ}
df-clab 1854(x {yφ} ↔ [x / y]φ)
df-cleq 1860(x(x yx z) → y = z)       (A = Bx(x Ax B))
df-clel 1863(A Bx(x = A x B))
wnfc 1990wff xA
df-nfc 1992(xAyx y A)
wne 2029wff AB
wnel 2030wff AB
df-ne 2031(AB ↔ ¬ A = B)
df-nel 2032(AB ↔ ¬ A B)
wral 2119wff x A φ
wrex 2120wff x A φ
wreu 2121wff ∃!x A φ
wrmo 2122wff ∃*x Aφ
crab 2123class {x Aφ}
df-ral 2124(x A φx(x Aφ))
df-rex 2125(x A φx(x A φ))
df-reu 2126(∃!x A φ∃!x(x A φ))
df-rmo 2127(∃*x Aφ∃*x(x A φ))
df-rab 2128{x Aφ} = {x ∣ (x A φ)}
cvv 2361class V
df-v 2363V = {xx = x}
ax-3 2512((¬ φ → ¬ ψ) → (ψφ))
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