Theorem ralab2 2705

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 2311	. 2 |- ( A. x e. { y | ph } ps <-> A. x ( x e. { y | ph } -> ps ) )
2		nfsab1 2030	. . . 4 |- F/ y x e. { y | ph }
3		nfv 1421	. . . 4 |- F/ y ps
4	2, 3	nfim 1464	. . 3 |- F/ y ( x e. { y | ph } -> ps )
5		nfv 1421	. . 3 |- F/ x ( ph -> ch )
6		eleq1 2100	. . . . 5 |- ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
7		abid 2028	. . . . 5 |- ( y e. { y | ph } <-> ph )
8	6, 7	syl6bb 185	. . . 4 |- ( x = y -> ( x e. { y | ph } <-> ph ) )
9		ralab2.1	. . . 4 |- ( x = y -> ( ps <-> ch ) )
10	8, 9	imbi12d 223	. . 3 |- ( x = y -> ( ( x e. { y | ph } -> ps ) <-> ( ph -> ch ) ) )
11	4, 5, 10	cbval 1637	. 2 |- ( A. x ( x e. { y | ph } -> ps ) <-> A. y ( ph -> ch ) )
12	1, 11	bitri 173	1 |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   e. wcel 1393   {cab 2026   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311

This theorem is referenced by:  ralrab2  2706  ssintab  3632