Theorem sprid 41724

Description: Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
sprid	|- { p | E. a e. _V E. b e. _V p = { a , b } } = { p | E. a E. b p = { a , b } }

Proof of Theorem sprid

Step	Hyp	Ref	Expression
1		rexv 3220	. . 3 |- ( E. a e. _V E. b e. _V p = { a , b } <-> E. a E. b e. _V p = { a , b } )
2		rexv 3220	. . . 4 |- ( E. b e. _V p = { a , b } <-> E. b p = { a , b } )
3	2	exbii 1774	. . 3 |- ( E. a E. b e. _V p = { a , b } <-> E. a E. b p = { a , b } )
4	1, 3	bitri 264	. 2 |- ( E. a e. _V E. b e. _V p = { a , b } <-> E. a E. b p = { a , b } )
5	4	abbii 2739	1 |- { p | E. a e. _V E. b e. _V p = { a , b } } = { p | E. a E. b p = { a , b } }

Syntax hints:   = wceq 1483   E.wex 1704   {cab 2608   E.wrex 2913   _Vcvv 3200   {cpr 4179

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-rex 2918  df-v 3202

This theorem is referenced by: (None)