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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-3 7((¬ φ → ¬ ψ) → (ψφ))
ax-mp 8φ    &   (φψ)       ψ
wa 96wff (φ ψ)
wb 97wff (φψ)
ax-ia1 98((φ ψ) → φ)
ax-ia2 99((φ ψ) → ψ)
ax-ia3 100(φ → (ψ → (φ ψ)))
df-bi 109(((φψ) → ((φψ) (ψφ))) (((φψ) (ψφ)) → (φψ)))
ax-in1 526((φ → ¬ φ) → ¬ φ)
ax-in2 527φ → (φψ))
wo 605wff (φ ψ)
ax-io 606(((φ χ) → ψ) ↔ ((φψ) (χψ)))
w3o 881wff (φ ψ χ)
w3a 882wff (φ ψ χ)
df-3or 883((φ ψ χ) ↔ ((φ ψ) χ))
df-3an 884((φ ψ χ) ↔ ((φ ψ) χ))
wtru 1228wff
wfal 1229wff
df-tru 1231( ⊤ ↔ (φφ))
df-fal 1232( ⊥ ↔ ¬ ⊤ )
wal 1253wff xφ
ax-5 1254(x(φψ) → (xφxψ))
ax-6 1255xφx ¬ xφ)
ax-7 1256(xyφyxφ)
ax-gen 1257φ       xφ
wex 1292wff xφ
ax-ie1 1293(xφxxφ)
ax-ie2 1294(x(ψxψ) → (x(φψ) ↔ (xφψ)))
cv 1300class x
wceq 1301wff A = B
wcel 1303wff A B
ax-8 1305(x = y → (x = zy = z))
ax-10 1306(x x = yy y = x)
ax-11 1307(x = y → (yφx(x = yφ)))
ax-i11e 1308(x = y → (yφx(x = y φ)))
ax-i12 1309(z z = x (z z = y z(x = yz x = y)))
ax-4 1310(xφφ)
ax-13 1313(x = y → (x zy z))
ax-14 1314(x = y → (z xz y))
ax-17 1319(φxφ)
ax-i9 1321x x = y
ax-5o 1328(x(xφψ) → (xφxψ))
ax-6o 1331x ¬ xφφ)
ax-ial 1333(xφxxφ)
ax-i5r 1334((xφxψ) → x(xφψ))
ax-9o 1438(x(x = yxφ) → φ)
ax-10o 1455(x x = y → (xφyφ))
wsbc 1483wff [A / x]φ
df-sb 1485([y / x]φ ↔ ((x = yφ) x(x = y φ)))
ax-16 1523(x x = y → (φxφ))
ax-11o 1533x x = y → (x = y → (φx(x = yφ))))
ax-15 1686z z = x → (¬ z z = y → (x yz x y)))
weu 1690wff ∃!xφ
wmo 1691wff ∃*xφ
df-eu 1694(∃!xφyx(φx = y))
df-mo 1695(∃*xφ ↔ (xφ∃!xφ))
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