Theorem ralrab2 3003

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- (x = y -> (ps <-> ch))

Assertion

Ref	Expression
ralrab2	|- (A.x e. {y e. A | ph}ps <-> A.y e. A (ph -> ch))

Distinct variable groups:   x,y   x,A   ch,x   ph,x   ps,y

Allowed substitution hints:   ph(y)   ps(x)   ch(y)   A(y)

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 |- {y e. A | ph} = {y | (y e. A /\ ph)}
2	1	raleqi 2812	. 2 |- (A.x e. {y e. A | ph}ps <-> A.x e. {y | (y e. A /\ ph)}ps)
3		ralab2.1	. . 3 |- (x = y -> (ps <-> ch))
4	3	ralab2 3002	. 2 |- (A.x e. {y | (y e. A /\ ph)}ps <-> A.y((y e. A /\ ph) -> ch))
5		impexp 433	. . . 4 |- (((y e. A /\ ph) -> ch) <-> (y e. A -> (ph -> ch)))
6	5	albii 1566	. . 3 |- (A.y((y e. A /\ ph) -> ch) <-> A.y(y e. A -> (ph -> ch)))
7		df-ral 2620	. . 3 |- (A.y e. A (ph -> ch) <-> A.y(y e. A -> (ph -> ch)))
8	6, 7	bitr4i 243	. 2 |- (A.y((y e. A /\ ph) -> ch) <-> A.y e. A (ph -> ch))
9	2, 4, 8	3bitri 262	1 |- (A.x e. {y e. A | ph}ps <-> A.y e. A (ph -> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  {cab 2339  A.wral 2615  {crab 2619

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-or 359  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-nfc 2479  df-ral 2620  df-rab 2624

This theorem is referenced by: (None)