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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 1187wff
wfal 1188wff
df-tru 1190( ⊤ ↔ (φφ))
df-fal 1191( ⊥ ↔ ¬ ⊤ )
wxo 1206wff (φψ)
df-xor 1207((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
wal 1266wff xφ
ax-5 1267(x(φψ) → (xφxψ))
ax-7 1268(xyφyxφ)
ax-gen 1269φ       xφ
wnf 1280wff xφ
df-nf 1281(Ⅎxφx(φxφ))
wex 1313wff xφ
ax-ie1 1314(xφxxφ)
ax-ie2 1315(x(ψxψ) → (x(φψ) ↔ (xφψ)))
cv 1323class x
wceq 1324wff A = B
wcel 1326wff A B
ax-8 1328(x = y → (x = zy = z))
ax-10 1329(x x = yy y = x)
ax-11 1330(x = y → (yφx(x = yφ)))
ax-i12 1331(z z = x (z z = y z(x = yz x = y)))
ax-bnd 1332(z z = x (z z = y xz(x = yz x = y)))
ax-4 1333(xφφ)
ax-13 1337(x = y → (x zy z))
ax-14 1338(x = y → (z xz y))
ax-17 1350(φxφ)
ax-i9 1354x x = y
ax-ial 1359(xφxxφ)
ax-i5r 1360((xφxψ) → x(xφψ))
ax-10o 1494(x x = y → (xφyφ))
wsbc 1531wff [A / x]φ
df-sb 1533([y / x]φ ↔ ((x = yφ) x(x = y φ)))
ax-16 1581(x x = y → (φxφ))
ax-11o 1590x x = y → (x = y → (φx(x = yφ))))
weu 1777wff ∃!xφ
wmo 1778wff ∃*xφ
df-eu 1780(∃!xφyx(φx = y))
df-mo 1781(∃*xφ ↔ (xφ∃!xφ))
ax-ext 1793(z(z xz y) → x = y)
cab 1796class {xφ}
df-clab 1797(x {yφ} ↔ [x / y]φ)
df-cleq 1803(x(x yx z) → y = z)       (A = Bx(x Ax B))
df-clel 1806(A Bx(x = A x B))
wnfc 1933wff xA
df-nfc 1935(xAyx y A)
wne 1972wff AB
wnel 1973wff AB
df-ne 1974(AB ↔ ¬ A = B)
df-nel 1975(AB ↔ ¬ A B)
ax-3 2062((¬ φ → ¬ ψ) → (ψφ))
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