Theorem exan3 34062

Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)

Assertion

Ref	Expression
exan3	|- ( ( A e. V /\ B e. W ) -> ( E. u ( A e. [ u ] R /\ B e. [ u ] R ) <-> E. u ( u R A /\ u R B ) ) )

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

Allowed substitution hint:   R( u)

Proof of Theorem exan3

Step	Hyp	Ref	Expression
1		elecALTV 34030	. . . 4 |- ( ( u e. _V /\ A e. V ) -> ( A e. [ u ] R <-> u R A ) )
2	1	el2v1 33985	. . 3 |- ( A e. V -> ( A e. [ u ] R <-> u R A ) )
3		elecALTV 34030	. . . 4 |- ( ( u e. _V /\ B e. W ) -> ( B e. [ u ] R <-> u R B ) )
4	3	el2v1 33985	. . 3 |- ( B e. W -> ( B e. [ u ] R <-> u R B ) )
5	2, 4	bi2anan9 917	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. [ u ] R /\ B e. [ u ] R ) <-> ( u R A /\ u R B ) ) )
6	5	exbidv 1850	1 |- ( ( A e. V /\ B e. W ) -> ( E. u ( A e. [ u ] R /\ B e. [ u ] R ) <-> E. u ( u R A /\ u R B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   [cec 7740

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-eu 2474  df-mo 2475  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-sbc 3436  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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by: (None)