Theorem rexab 3369

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab.1	|- ( y = x -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexab	|- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Distinct variable groups:   x, y   ps, y

Allowed substitution hints:   ph( x, y)   ps( x)   ch( x, y)

Proof of Theorem rexab

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 |- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
2		vex 3203	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 3350	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	anbi1i 731	. . 3 |- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii 1774	. 2 |- ( E. x ( x e. { y | ph } /\ ch ) <-> E. x ( ps /\ ch ) )
7	1, 6	bitri 264	1 |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913

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-rex 2918  df-v 3202

This theorem is referenced by:  4sqlem12  15660  mblfinlem3  33448  mblfinlem4  33449  ismblfin  33450  itg2addnclem  33461  itg2addnc  33464  diophrex  37339