Theorem ralab2 3119

Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralab2	|- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )

Distinct variable groups:   x, y   ch, x   ph, x   ps, y

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

Proof of Theorem ralab2

Step	Hyp	Ref	Expression
1		df-ral 2715	. 2 |- ( A. x e. { y | ph } ps <-> A. x ( x e. { y | ph } -> ps ) )
2		nfsab1 2428	. . . 4 |- F/ y x e. { y | ph }
3		nfv 1673	. . . 4 |- F/ y ps
4	2, 3	nfim 1852	. . 3 |- F/ y ( x e. { y | ph } -> ps )
5		nfv 1673	. . 3 |- F/ x ( ph -> ch )
6		eleq1 2498	. . . . 5 |- ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
7		abid 2426	. . . . 5 |- ( y e. { y | ph } <-> ph )
8	6, 7	syl6bb 261	. . . 4 |- ( x = y -> ( x e. { y | ph } <-> ph ) )
9		ralab2.1	. . . 4 |- ( x = y -> ( ps <-> ch ) )
10	8, 9	imbi12d 320	. . 3 |- ( x = y -> ( ( x e. { y | ph } -> ps ) <-> ( ph -> ch ) ) )
11	4, 5, 10	cbval 1969	. 2 |- ( A. x ( x e. { y | ph } -> ps ) <-> A. y ( ph -> ch ) )
12	1, 11	bitri 249	1 |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1367   e. wcel 1756   {cab 2424   A.wral 2710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2715

This theorem is referenced by:  ralrab2  3120  ssintab  4140  efgval  16205  efger  16206