Theorem bj-sngleq 32955

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.

Step	Hyp	Ref	Expression
1		rexeq 3139	. . 3 |- ( A = B -> ( E. y e. A x = { y } <-> E. y e. B x = { y } ) )
2	1	abbidv 2741	. 2 |- ( A = B -> { x | E. y e. A x = { y } } = { x | E. y e. B x = { y } } )
3		df-bj-sngl 32954	. 2 |- sngl A = { x | E. y e. A x = { y } }
4		df-bj-sngl 32954	. 2 |- sngl B = { x | E. y e. B x = { y } }
5	2, 3, 4	3eqtr4g 2681	1 |- ( A = B -> sngl A = sngl B )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   {csn 4177  sngl bj-csngl 32953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-bj-sngl 32954

This theorem is referenced by:  bj-tageq  32964