Theorem mob2 3016

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 957	. . 3 |- ((A e. B /\ E*xph /\ ph) -> ph)
2		moi2.1	. . 3 |- (x = A -> (ph <-> ps))
3	1, 2	syl5ibcom 211	. 2 |- ((A e. B /\ E*xph /\ ph) -> (x = A -> ps))
4		nfs1v 2106	. . . . . . . 8 |- F/x[y / x]ph
5		sbequ12 1919	. . . . . . . 8 |- (x = y -> (ph <-> [y / x]ph))
6	4, 5	mo4f 2236	. . . . . . 7 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
7		sp 1747	. . . . . . 7 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> A.y((ph /\ [y / x]ph) -> x = y))
8	6, 7	sylbi 187	. . . . . 6 |- (E*xph -> A.y((ph /\ [y / x]ph) -> x = y))
9		nfv 1619	. . . . . . . . . 10 |- F/xps
10	9, 2	sbhypf 2904	. . . . . . . . 9 |- (y = A -> ([y / x]ph <-> ps))
11	10	anbi2d 684	. . . . . . . 8 |- (y = A -> ((ph /\ [y / x]ph) <-> (ph /\ ps)))
12		eqeq2 2362	. . . . . . . 8 |- (y = A -> (x = y <-> x = A))
13	11, 12	imbi12d 311	. . . . . . 7 |- (y = A -> (((ph /\ [y / x]ph) -> x = y) <-> ((ph /\ ps) -> x = A)))
14	13	spcgv 2939	. . . . . 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 418	. . . 4 |- ((A e. B /\ E*xph) -> ((ph /\ ps) -> x = A))
17	16	exp3a 425	. . 3 |- ((A e. B /\ E*xph) -> (ph -> (ps -> x = A)))
18	17	3impia 1148	. 2 |- ((A e. B /\ E*xph /\ ph) -> (ps -> x = A))
19	3, 18	impbid 183	1 |- ((A e. B /\ E*xph /\ ph) -> (x = A <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540   = wceq 1642  [wsb 1648   e. wcel 1710  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by:  moi2  3017  mob  3018