Theorem isgbe 41639

Description: The predicate "is an even Goldbach number". An even Goldbach number is an even integer having a Goldbach partition, i.e. which can be written as a sum of two odd primes. (Contributed by AV, 20-Jul-2020.)

Assertion

Ref	Expression
isgbe	|- ( Z e. GoldbachEven <-> ( Z e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )

Distinct variable group:   Z, p, q

Proof of Theorem isgbe

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . 4 |- ( z = Z -> ( z = ( p + q ) <-> Z = ( p + q ) ) )
2	1	3anbi3d 1405	. . 3 |- ( z = Z -> ( ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) <-> ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )
3	2	2rexbidv 3057	. 2 |- ( z = Z -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) <-> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )
4		df-gbe 41636	. 2 |- GoldbachEven = { z e. Even | E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) }
5	3, 4	elrab2 3366	1 |- ( Z e. GoldbachEven <-> ( Z e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   + caddc 9939   Primecprime 15385   Even ceven 41537   Odd codd 41538   GoldbachEven cgbe 41633

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-3an 1039  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-rab 2921  df-v 3202  df-gbe 41636

This theorem is referenced by:  gbeeven  41642  gbepos  41646  gbegt5  41649  6gbe  41659  8gbe  41661  sbgoldbwt  41665  sbgoldbst  41666  sbgoldbalt  41669  nnsum3primesgbe  41680  bgoldbtbndlem4  41696  bgoldbtbnd  41697