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Theorem nfeqf 1958
 Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-12o 2142. (Contributed by Mario Carneiro, 6-Oct-2016.)
Assertion
Ref Expression
nfeqf ((¬ z z = x ¬ z z = y) → Ⅎz x = y)

Proof of Theorem nfeqf
StepHypRef Expression
1 nfnae 1956 . . 3 z ¬ z z = x
2 nfnae 1956 . . 3 z ¬ z z = y
31, 2nfan 1824 . 2 zz z = x ¬ z z = y)
4 ax12o 1934 . . 3 z z = x → (¬ z z = y → (x = yz x = y)))
54imp 418 . 2 ((¬ z z = x ¬ z z = y) → (x = yz x = y))
63, 5nfd 1766 1 ((¬ z z = x ¬ z z = y) → Ⅎz x = y)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  equvini  1987  equveli  1988  nfsb4t  2080  sbcom  2089  nfeud2  2216
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