Theorem rabsnt 2900

Description: Truth implied by equality of a restricted class abstraction and a singleton.

Hypotheses

Ref	Expression
rabsnt.1	|- B e. V
rabsnt.2	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
rabsnt	|- ({x e. A | ph} = {B} -> ps)

Distinct variable groups:   x,A   x,B   ps,x

Proof of Theorem rabsnt

Step	Hyp	Ref	Expression
1		rabsnt.2	. . . 4 |- (x = B -> (ph <-> ps))
2	1	reuuni2 2890	. . 3 |- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))
3	2	biimprd 154	. 2 |- ((B e. A /\ E!x e. A ph) -> (U.{x e. A | ph} = B -> ps))
4		ssrab2 2134	. . . . 5 |- {x e. A | ph} (_ A
5		sseq1 2085	. . . . 5 |- ({x e. A | ph} = {B} -> ({x e. A | ph} (_ A <-> {B} (_ A))
6	4, 5	mpbii 193	. . . 4 |- ({x e. A | ph} = {B} -> {B} (_ A)
7		rabsnt.1	. . . . 5 |- B e. V
8	7	snss 2465	. . . 4 |- (B e. A <-> {B} (_ A)
9	6, 8	sylibr 200	. . 3 |- ({x e. A | ph} = {B} -> B e. A)
10	7	reusni 2899	. . 3 |- ({x e. A | ph} = {B} -> E!x e. A ph)
11	9, 10	jca 288	. 2 |- ({x e. A | ph} = {B} -> (B e. A /\ E!x e. A ph))
12		unieq 2514	. . 3 |- ({x e. A | ph} = {B} -> U.{x e. A | ph} = U.{B})
13	7	unisn 2521	. . 3 |- U.{B} = B
14	12, 13	syl6eq 1526	. 2 |- ({x e. A | ph} = {B} -> U.{x e. A | ph} = B)
15	3, 11, 14	sylc 68	1 |- ({x e. A | ph} = {B} -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E!wreu 1650  {crab 1651  Vcvv 1814   (_ wss 2050  {csn 2413  U.cuni 2507

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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