Theorem rabxfr 2902

Description: Abstraction class membership after substituting an expression A (containing y) for x in the class expression ph.

Hypotheses

Ref	Expression
rabxfr.1	|- (z e. B -> A.y z e. B)
rabxfr.2	|- (z e. C -> A.y z e. C)
rabxfr.3	|- (y e. D -> A e. D)
rabxfr.4	|- (x = A -> (ph <-> ps))
rabxfr.5	|- (y = B -> A = C)

Assertion

Ref	Expression
rabxfr	|- (B e. D -> (C e. {x e. D | ph} <-> B e. {y e. D | ps}))

Distinct variable groups:   x,A   z,B   z,C   x,y,z,D   ph,y,z   ps,x,z

Proof of Theorem rabxfr

Step	Hyp	Ref	Expression
1		rabxfr.3	. . . . . . . 8 |- (y e. D -> A e. D)
2		ibibr 591	. . . . . . . 8 |- ((y e. D -> A e. D) <-> (y e. D -> (A e. D <-> y e. D)))
3	1, 2	mpbi 189	. . . . . . 7 |- (y e. D -> (A e. D <-> y e. D))
4	3	anbi1d 617	. . . . . 6 |- (y e. D -> ((A e. D /\ ps) <-> (y e. D /\ ps)))
5		rabxfr.4	. . . . . . 7 |- (x = A -> (ph <-> ps))
6	5	elrab 1905	. . . . . 6 |- (A e. {x e. D | ph} <-> (A e. D /\ ps))
7		rabid 1769	. . . . . 6 |- (y e. {y e. D | ps} <-> (y e. D /\ ps))
8	4, 6, 7	3bitr4g 555	. . . . 5 |- (y e. D -> (A e. {x e. D | ph} <-> y e. {y e. D | ps}))
9	8	rabbii 1805	. . . 4 |- {y e. D | A e. {x e. D | ph}} = {y e. D | y e. {y e. D | ps}}
10	9	eleq2i 1538	. . 3 |- (B e. {y e. D | A e. {x e. D | ph}} <-> B e. {y e. D | y e. {y e. D | ps}})
11		rabxfr.1	. . . 4 |- (z e. B -> A.y z e. B)
12		ax-17 971	. . . 4 |- (z e. D -> A.y z e. D)
13		rabxfr.2	. . . . 5 |- (z e. C -> A.y z e. C)
14		ax-17 971	. . . . 5 |- (z e. {x e. D | ph} -> A.y z e. {x e. D | ph})
15	13, 14	hbel 1566	. . . 4 |- (C e. {x e. D | ph} -> A.y C e. {x e. D | ph})
16		rabxfr.5	. . . . 5 |- (y = B -> A = C)
17	16	eleq1d 1540	. . . 4 |- (y = B -> (A e. {x e. D | ph} <-> C e. {x e. D | ph}))
18	11, 12, 15, 17	elrabf 1904	. . 3 |- (B e. {y e. D | A e. {x e. D | ph}} <-> (B e. D /\ C e. {x e. D | ph}))
19		hbrab1 1772	. . . . 5 |- (z e. {y e. D | ps} -> A.y z e. {y e. D | ps})
20	11, 19	hbel 1566	. . . 4 |- (B e. {y e. D | ps} -> A.y B e. {y e. D | ps})
21		eleq1 1534	. . . 4 |- (y = B -> (y e. {y e. D | ps} <-> B e. {y e. D | ps}))
22	11, 12, 20, 21	elrabf 1904	. . 3 |- (B e. {y e. D | y e. {y e. D | ps}} <-> (B e. D /\ B e. {y e. D | ps}))
23	10, 18, 22	3bitr3 181	. 2 |- ((B e. D /\ C e. {x e. D | ph}) <-> (B e. D /\ B e. {y e. D | ps}))
24		pm5.32 644	. 2 |- ((B e. D -> (C e. {x e. D | ph} <-> B e. {y e. D | ps})) <-> ((B e. D /\ C e. {x e. D | ph}) <-> (B e. D /\ B e. {y e. D | ps})))
25	23, 24	mpbir 190	1 |- (B e. D -> (C e. {x e. D | ph} <-> B e. {y e. D | ps}))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  {crab 1648

This theorem is referenced by:  reuunixfr 2906  dfuz 6202

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812

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