Theorem ralab2 3224

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 2800	. 2 |- ( A. x e. { y | ph } ps <-> A. x ( x e. { y | ph } -> ps ) )
2		nfsab1 2440	. . . 4 |- F/ y x e. { y | ph }
3		nfv 1674	. . . 4 |- F/ y ps
4	2, 3	nfim 1855	. . 3 |- F/ y ( x e. { y | ph } -> ps )
5		nfv 1674	. . 3 |- F/ x ( ph -> ch )
6		eleq1 2523	. . . . 5 |- ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
7		abid 2438	. . . . 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 1978	. 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 1368   e. wcel 1758   {cab 2436   A.wral 2795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ral 2800

This theorem is referenced by:  ralrab2  3225  ssintab  4246  efgval  16327  efger  16328