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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 860wff (φ ψ χ)
w3a 861wff (φ ψ χ)
df-3or 862((φ ψ χ) ↔ ((φ ψ) χ))
df-3an 863((φ ψ χ) ↔ ((φ ψ) χ))
wtru 1206wff
wfal 1207wff
df-tru 1209( ⊤ ↔ (φφ))
df-fal 1210( ⊥ ↔ ¬ ⊤ )
wal 1231wff xφ
ax-5 1232(x(φψ) → (xφxψ))
ax-6 1233xφx ¬ xφ)
ax-7 1234(xyφyxφ)
ax-gen 1235φ       xφ
wex 1270wff xφ
ax-ie1 1271(xφxxφ)
ax-ie2 1272(x(ψxψ) → (x(φψ) ↔ (xφψ)))
cv 1278class x
wceq 1279wff A = B
wcel 1281wff A B
ax-8 1283(x = y → (x = zy = z))
ax-10 1284(x x = yy y = x)
ax-11 1285(x = y → (yφx(x = yφ)))
ax-i11e 1286(x = y → (yφx(x = y φ)))
ax-i12 1287(z z = x (z z = y z(x = yz x = y)))
ax-4 1288(xφφ)
ax-13 1291(x = y → (x zy z))
ax-14 1292(x = y → (z xz y))
ax-17 1297(φxφ)
ax-i9 1299x x = y
ax-5o 1305(x(xφψ) → (xφxψ))
ax-6o 1308x ¬ xφφ)
ax-ial 1310(xφxxφ)
ax-i5r 1311((xφxψ) → x(xφψ))
ax-9o 1411(x(x = yxφ) → φ)
ax-10o 1427(x x = y → (xφyφ))
wsbc 1457wff [A / x]φ
df-sb 1459([y / x]φ ↔ ((x = yφ) x(x = y φ)))
ax-16 1502(x x = y → (φxφ))
ax-11o 1512x x = y → (x = y → (φx(x = yφ))))
ax-15 1681z z = x → (¬ z z = y → (x yz x y)))
weu 1688wff ∃!xφ
wmo 1689wff ∃*xφ
df-eu 1692(∃!xφyx(φx = y))
df-mo 1693(∃*xφ ↔ (xφ∃!xφ))
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