Theorem grothprimlem 9655

Description: Lemma for grothprim 9656. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)

Assertion

Ref	Expression
grothprimlem	|- ( { u , v } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )

Distinct variable group:   w, v, u, h, g

Proof of Theorem grothprimlem

Step	Hyp	Ref	Expression
1		dfpr2 4195	. . 3 |- { u , v } = { h | ( h = u \/ h = v ) }
2	1	eleq1i 2692	. 2 |- ( { u , v } e. w <-> { h | ( h = u \/ h = v ) } e. w )
3		clabel 2749	. 2 |- ( { h | ( h = u \/ h = v ) } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )
4	2, 3	bitri 264	1 |- ( { u , v } e. w <-> E. g ( g e. w /\ A. h ( h e. g <-> ( h = u \/ h = v ) ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   {cab 2608   {cpr 4179

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-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  grothprim  9656