Theorem mob2 2715

Description: Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)

Hypothesis

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

Assertion

Ref	Expression
mob2	|- ((A e. B /\ E*xph /\ ph) -> (x = A <-> ps))

Distinct variable groups:   x,A   ps,x

Allowed substitution hints:   ph(x)   B(x)

Proof of Theorem mob2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 905	. . 3 |- ((A e. B /\ E*xph /\ ph) -> ph)
2		moi2.1	. . 3 |- (x = A -> (ph <-> ps))
3	1, 2	syl5ibcom 144	. 2 |- ((A e. B /\ E*xph /\ ph) -> (x = A -> ps))
4		nfs1v 1812	. . . . . . . 8 |- F/x[y / x]ph
5		sbequ12 1651	. . . . . . . 8 |- (x = y -> (ph <-> [y / x]ph))
6	4, 5	mo4f 1957	. . . . . . 7 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
7		sp 1398	. . . . . . 7 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> A.y((ph /\ [y / x]ph) -> x = y))
8	6, 7	sylbi 114	. . . . . 6 |- (E*xph -> A.y((ph /\ [y / x]ph) -> x = y))
9		nfv 1418	. . . . . . . . . 10 |- F/xps
10	9, 2	sbhypf 2597	. . . . . . . . 9 |- (y = A -> ([y / x]ph <-> ps))
11	10	anbi2d 437	. . . . . . . 8 |- (y = A -> ((ph /\ [y / x]ph) <-> (ph /\ ps)))
12		eqeq2 2046	. . . . . . . 8 |- (y = A -> (x = y <-> x = A))
13	11, 12	imbi12d 223	. . . . . . 7 |- (y = A -> (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = A)))
14	13	spcgv 2634	. . . . . 6 |- (A e. B -> (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ ps) -> x = A)))
15	8, 14	syl5 28	. . . . 5 |- (A e. B -> (E*xph -> ((ph /\ ps) -> x = A)))
16	15	imp 115	. . . 4 |- ((A e. B /\ E*xph) -> ((ph /\ ps) -> x = A))
17	16	expd 245	. . 3 |- ((A e. B /\ E*xph) -> (ph -> (ps -> x = A)))
18	17	3impia 1100	. 2 |- ((A e. B /\ E*xph /\ ph) -> (ps -> x = A))
19	3, 18	impbid 120	1 |- ((A e. B /\ E*xph /\ ph) -> (x = A <-> ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884  A.wal 1240   = wceq 1242   e. wcel 1390  [wsb 1642  E*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  moi2  2716  mob  2717