Theorem ralab2 2928

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 2480	. 2 |- ( A. x e. { y | ph } ps <-> A. x ( x e. { y | ph } -> ps ) )
2		nfsab1 2186	. . . 4 |- F/ y x e. { y | ph }
3		nfv 1542	. . . 4 |- F/ y ps
4	2, 3	nfim 1586	. . 3 |- F/ y ( x e. { y | ph } -> ps )
5		nfv 1542	. . 3 |- F/ x ( ph -> ch )
6		eleq1 2259	. . . . 5 |- ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
7		abid 2184	. . . . 5 |- ( y e. { y | ph } <-> ph )
8	6, 7	bitrdi 196	. . . 4 |- ( x = y -> ( x e. { y | ph } <-> ph ) )
9		ralab2.1	. . . 4 |- ( x = y -> ( ps <-> ch ) )
10	8, 9	imbi12d 234	. . 3 |- ( x = y -> ( ( x e. { y | ph } -> ps ) <-> ( ph -> ch ) ) )
11	4, 5, 10	cbval 1768	. 2 |- ( A. x ( x e. { y | ph } -> ps ) <-> A. y ( ph -> ch ) )
12	1, 11	bitri 184	1 |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   e. wcel 2167   {cab 2182   A.wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480

This theorem is referenced by:  ralrab2  2929  ssintab  3891