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Theorem axc16nf 2027
Description: If dtru 4594 is false, then there is only one element in the universe, so everything satisfies  F/. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 1920. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.)
Assertion
Ref Expression
axc16nf  |-  ( A. x  x  =  y  ->  F/ z ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem axc16nf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 aev 2026 . 2  |-  ( A. x  x  =  y  ->  A. z  z  =  w )
2 nfa1 1979 . . 3  |-  F/ z A. z  z  =  w
3 axc16 2024 . . 3  |-  ( A. z  z  =  w  ->  ( ph  ->  A. z ph ) )
42, 3nfd 1956 . 2  |-  ( A. z  z  =  w  ->  F/ z ph )
51, 4syl 17 1  |-  ( A. x  x  =  y  ->  F/ z ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1442   F/wnf 1667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  nfsb  2269  nfsbd  2271
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