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Theorem a16nf 1743
 Description: If there is only one element in the universe, then everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.)
Assertion
Ref Expression
a16nf (x x = y → Ⅎzφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem a16nf
StepHypRef Expression
1 nfae 1604 . 2 zx x = y
2 a16g 1741 . 2 (x x = y → (φzφ))
31, 2nfd 1413 1 (x x = y → Ⅎzφ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240  Ⅎwnf 1346 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  nfsbxy  1815  nfsbxyt  1816  dvelimor  1891
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