Theorem prtlem9 34149

Description: Lemma for prter3 34167. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Assertion

Ref	Expression
prtlem9	|- ( A e. B -> E. x e. B [ x ] .~ = [ A ] .~ )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   .~ ( x)

Proof of Theorem prtlem9

Step	Hyp	Ref	Expression
1		risset 3062	. 2 |- ( A e. B <-> E. x e. B x = A )
2		eceq1 7782	. . 3 |- ( x = A -> [ x ] .~ = [ A ] .~ )
3	2	reximi 3011	. 2 |- ( E. x e. B x = A -> E. x e. B [ x ] .~ = [ A ] .~ )
4	1, 3	sylbi 207	1 |- ( A e. B -> E. x e. B [ x ] .~ = [ A ] .~ )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   [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

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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by: (None)