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Theorem bj-sngleq 33606
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 3059 . . 3  |-  ( A  =  B  ->  ( E. y  e.  A  x  =  { y } 
<->  E. y  e.  B  x  =  { y } ) )
21abbidv 2603 . 2  |-  ( A  =  B  ->  { x  |  E. y  e.  A  x  =  { y } }  =  {
x  |  E. y  e.  B  x  =  { y } }
)
3 df-bj-sngl 33605 . 2  |- sngl  A  =  { x  |  E. y  e.  A  x  =  { y } }
4 df-bj-sngl 33605 . 2  |- sngl  B  =  { x  |  E. y  e.  B  x  =  { y } }
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  -> sngl  A  = sngl 
B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   {cab 2452   E.wrex 2815   {csn 4027  sngl bj-csngl 33604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-bj-sngl 33605
This theorem is referenced by:  bj-tageq  33615
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