Theorem moi2 1924

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 1813	. . . . . . . . 9 |- y e. V
2	1	eqvinc 1883	. . . . . . . 8 |- (y = A <-> E.x(x = y /\ x = A))
3		hbs1 1332	. . . . . . . . . 10 |- ([y / x]ph -> A.x[y / x]ph)
4		ax-17 971	. . . . . . . . . 10 |- (ps -> A.xps)
5	3, 4	hbbi 1010	. . . . . . . . 9 |- (([y / x]ph <-> ps) -> A.x([y / x]ph <-> ps))
6		sbequ12 1181	. . . . . . . . . . 11 |- (x = y -> (ph <-> [y / x]ph))
7	6	bicomd 521	. . . . . . . . . 10 |- (x = y -> ([y / x]ph <-> ph))
8		moi2.1	. . . . . . . . . 10 |- (x = A -> (ph <-> ps))
9	7, 8	sylan9bb 540	. . . . . . . . 9 |- ((x = y /\ x = A) -> ([y / x]ph <-> ps))
10	5, 9	19.23ai 1064	. . . . . . . 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 616	. . . . . 6 |- (y = A -> ((ph /\ [y / x]ph) <-> (ph /\ ps)))
13		eqeq2 1484	. . . . . 6 |- (y = A -> (x = y <-> x = A))
14	12, 13	imbi12d 626	. . . . 5 |- (y = A -> (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = A)))
15	14	cla4gv 1862	. . . 4 |- (A e. B -> (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ ps) -> x = A)))
16	15	a4sd 985	. . 3 |- (A e. B -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ ps) -> x = A)))
17	3, 6	mo4f 1402	. . 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 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  E*wmo 1381

This theorem is referenced by:  spwpr3OLD 8662

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-9 965  ax-10 966  ax-11 967  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-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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