Theorem exanres 34063

Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.)

Assertion

Ref	Expression
exanres	|- ( ( B e. V /\ C e. W ) -> ( E. u ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> E. u e. A ( u R B /\ u S C ) ) )

Distinct variable groups:   u, B   u, C   u, V   u, W

Allowed substitution hints:   A( u)   R( u)   S( u)

Proof of Theorem exanres

Step	Hyp	Ref	Expression
1		brresALTV 34032	. . . . 5 |- ( B e. V -> ( u ( R |` A ) B <-> ( u e. A /\ u R B ) ) )
2		brresALTV 34032	. . . . 5 |- ( C e. W -> ( u ( S |` A ) C <-> ( u e. A /\ u S C ) ) )
3	1, 2	bi2anan9 917	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> ( ( u e. A /\ u R B ) /\ ( u e. A /\ u S C ) ) ) )
4		anandi 871	. . . 4 |- ( ( u e. A /\ ( u R B /\ u S C ) ) <-> ( ( u e. A /\ u R B ) /\ ( u e. A /\ u S C ) ) )
5	3, 4	syl6bbr 278	. . 3 |- ( ( B e. V /\ C e. W ) -> ( ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> ( u e. A /\ ( u R B /\ u S C ) ) ) )
6	5	exbidv 1850	. 2 |- ( ( B e. V /\ C e. W ) -> ( E. u ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> E. u ( u e. A /\ ( u R B /\ u S C ) ) ) )
7		df-rex 2918	. 2 |- ( E. u e. A ( u R B /\ u S C ) <-> E. u ( u e. A /\ ( u R B /\ u S C ) ) )
8	6, 7	syl6bbr 278	1 |- ( ( B e. V /\ C e. W ) -> ( E. u ( u ( R |` A ) B /\ u ( S |` A ) C ) <-> E. u e. A ( u R B /\ u S C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E.wrex 2913   class class class wbr 4653   |` cres 5116

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  exanres2  34065