Theorem isgbo 41641

Description: The predicate "is an odd Goldbach number". An odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020.)

Assertion

Ref	Expression
isgbo	|- ( Z e. GoldbachOdd <-> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )

Distinct variable group:   Z, p, q, r

Proof of Theorem isgbo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . 5 |- ( z = Z -> ( z = ( ( p + q ) + r ) <-> Z = ( ( p + q ) + r ) ) )
2	1	anbi2d 740	. . . 4 |- ( z = Z -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
3	2	rexbidv 3052	. . 3 |- ( z = Z -> ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
4	3	2rexbidv 3057	. 2 |- ( z = Z -> ( E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
5		df-gbo 41638	. 2 |- GoldbachOdd = { z e. Odd | E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) }
6	4, 5	elrab2 3366	1 |- ( Z e. GoldbachOdd <-> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )

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   Odd codd 41538   GoldbachOdd cgbo 41635

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-rab 2921  df-v 3202  df-gbo 41638

This theorem is referenced by:  gbogbow  41644  gboge9  41652  9gbo  41662  11gbo  41663  sbgoldbst  41666  nnsum4primesoddALTV  41685  bgoldbtbnd  41697