Theorem gbogbow 41644

Description: A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.)

Assertion

Ref	Expression
gbogbow	|- ( Z e. GoldbachOdd -> Z e. GoldbachOddW )

Proof of Theorem gbogbow

Dummy variables p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) -> Z = ( ( p + q ) + r ) )
2	1	reximi 3011	. . . . 5 |- ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) -> E. r e. Prime Z = ( ( p + q ) + r ) )
3	2	reximi 3011	. . . 4 |- ( E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) -> E. q e. Prime E. r e. Prime Z = ( ( p + q ) + r ) )
4	3	reximi 3011	. . 3 |- ( 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 Z = ( ( p + q ) + r ) )
5	4	anim2i 593	. 2 |- ( ( 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 ) ) ) -> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime Z = ( ( p + q ) + r ) ) )
6		isgbo 41641	. 2 |- ( 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 ) ) ) )
7		isgbow 41640	. 2 |- ( Z e. GoldbachOddW <-> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime Z = ( ( p + q ) + r ) ) )
8	5, 6, 7	3imtr4i 281	1 |- ( Z e. GoldbachOdd -> Z e. GoldbachOddW )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   + caddc 9939   Primecprime 15385   Odd codd 41538   GoldbachOddW cgbow 41634   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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-gbow 41637  df-gbo 41638

This theorem is referenced by:  gboodd  41645  gbopos  41648  stgoldbwt  41664