Theorem moi2 1927

Description: Consequence of "at most one."

Hypothesis

Ref	Expression
moi2.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
moi2	|- (((A e. B /\ E*xph) /\ (ph /\ ps)) -> x = A)

Distinct variable groups:   x,A   ps,x

Proof of Theorem moi2

Step	Hyp	Ref	Expression
1		visset 1816	. . . . . . . . 9 |- y e. V
2	1	eqvinc 1886	. . . . . . . 8 |- (y = A <-> E.x(x = y /\ x = A))
3		hbs1 1334	. . . . . . . . . 10 |- ([y / x]ph -> A.x[y / x]ph)
4		ax-17 973	. . . . . . . . . 10 |- (ps -> A.xps)
5	3, 4	hbbi 1012	. . . . . . . . 9 |- (([y / x]ph <-> ps) -> A.x([y / x]ph <-> ps))
6		sbequ12 1183	. . . . . . . . . . 11 |- (x = y -> (ph <-> [y / x]ph))
7	6	bicomd 523	. . . . . . . . . 10 |- (x = y -> ([y / x]ph <-> ph))
8		moi2.1	. . . . . . . . . 10 |- (x = A -> (ph <-> ps))
9	7, 8	sylan9bb 542	. . . . . . . . 9 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
10	5, 9	19.23ai 1066	. . . . . . . 8 |- (E.x(x = y /\ x = A) -> ([y / x]ph <-> ps))
11	2, 10	sylbi 199	. . . . . . 7 |- (y = A -> ([y / x]ph <-> ps))
12	11	anbi2d 618	. . . . . 6 |- (y = A -> ((ph /\ [y / x]ph) <-> (ph /\ ps)))
13		eqeq2 1487	. . . . . 6 |- (y = A -> (x = y <-> x = A))
14	12, 13	imbi12d 628	. . . . 5 |- (y = A -> (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = A)))
15	14	cla4gv 1865	. . . 4 |- (A e. B -> (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ ps) -> x = A)))
16	15	a4sd 987	. . 3 |- (A e. B -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ ps) -> x = A)))
17	3, 6	mo4f 1404	. . 3 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
18	16, 17	syl5ib 206	. 2 |- (A e. B -> (E*xph -> ((ph /\ ps) -> x = A)))
19	18	imp31 362	1 |- (((A e. B /\ E*xph) /\ (ph /\ ps)) -> x = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  E*wmo 1383

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-9 967  ax-10 968  ax-11 969  ax-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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