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Theorem bj-sngleq 31077
Description: Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sngleq  |-  ( A  =  B  -> sngl  A  = sngl 
B )

Proof of Theorem bj-sngleq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 3004 . . 3  |-  ( A  =  B  ->  ( E. y  e.  A  x  =  { y } 
<->  E. y  e.  B  x  =  { y } ) )
21abbidv 2538 . 2  |-  ( A  =  B  ->  { x  |  E. y  e.  A  x  =  { y } }  =  {
x  |  E. y  e.  B  x  =  { y } }
)
3 df-bj-sngl 31076 . 2  |- sngl  A  =  { x  |  E. y  e.  A  x  =  { y } }
4 df-bj-sngl 31076 . 2  |- sngl  B  =  { x  |  E. y  e.  B  x  =  { y } }
52, 3, 43eqtr4g 2468 1  |-  ( A  =  B  -> sngl  A  = sngl 
B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   {cab 2387   E.wrex 2754   {csn 3971  sngl bj-csngl 31075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-bj-sngl 31076
This theorem is referenced by:  bj-tageq  31086
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