Theorem ralab2 3002

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 2620	. 2 |- (A.x e. {y | ph}ps <-> A.x(x e. {y | ph} -> ps))
2		nfsab1 2343	. . . 4 |- F/y x e. {y | ph}
3		nfv 1619	. . . 4 |- F/yps
4	2, 3	nfim 1813	. . 3 |- F/y(x e. {y | ph} -> ps)
5		nfv 1619	. . 3 |- F/x(ph -> ch)
6		eleq1 2413	. . . . 5 |- (x = y -> (x e. {y | ph} <-> y e. {y | ph}))
7		abid 2341	. . . . 5 |- (y e. {y | ph} <-> ph)
8	6, 7	syl6bb 252	. . . 4 |- (x = y -> (x e. {y | ph} <-> ph))
9		ralab2.1	. . . 4 |- (x = y -> (ps <-> ch))
10	8, 9	imbi12d 311	. . 3 |- (x = y -> ((x e. {y | ph} -> ps) <-> (ph -> ch)))
11	4, 5, 10	cbval 1984	. 2 |- (A.x(x e. {y | ph} -> ps) <-> A.y(ph -> ch))
12	1, 11	bitri 240	1 |- (A.x e. {y | ph}ps <-> A.y(ph -> ch))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620

This theorem is referenced by:  ralrab2  3003  ssintab  3944