Theorem ralab2 2699

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 2305	. 2 |- (A.x e. {y | ph }ps <-> A.x(x e. {y | ph } -> ps))
2		nfsab1 2027	. . . 4 |- F/y x e. {y | ph }
3		nfv 1418	. . . 4 |- F/yps
4	2, 3	nfim 1461	. . 3 |- F/y(x e. {y | ph } -> ps)
5		nfv 1418	. . 3 |- F/x(ph -> ch)
6		eleq1 2097	. . . . 5 |- (x = y -> (x e. {y | ph } <-> y e. {y | ph }))
7		abid 2025	. . . . 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 1634	. 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 1240   e. wcel 1390   {cab 2023  A.wral 2300

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305

This theorem is referenced by:  ralrab2  2700  ssintab  3623