Theorem mosubt 2712

Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)

Assertion

Ref	Expression
mosubt	|- (A.yE*xph -> E*xE.y(y = A /\ ph))

Distinct variable group:   x,y,A

Allowed substitution hints:   ph(x,y)

Proof of Theorem mosubt

Step	Hyp	Ref	Expression
1		eueq 2706	. . . . . 6 |- (A e. _V <-> E!y y = A)
2		isset 2555	. . . . . 6 |- (A e. _V <-> E.y y = A)
3	1, 2	bitr3i 175	. . . . 5 |- (E!y y = A <-> E.y y = A)
4		nfv 1418	. . . . . 6 |- F/x y = A
5	4	euexex 1982	. . . . 5 |- ((E!y y = A /\ A.yE*xph) -> E*xE.y(y = A /\ ph))
6	3, 5	sylanbr 269	. . . 4 |- ((E.y y = A /\ A.yE*xph) -> E*xE.y(y = A /\ ph))
7	6	expcom 109	. . 3 |- (A.yE*xph -> (E.y y = A -> E*xE.y(y = A /\ ph)))
8		moanimv 1972	. . 3 |- (E*x(E.y y = A /\ E.y(y = A /\ ph)) <-> (E.y y = A -> E*xE.y(y = A /\ ph)))
9	7, 8	sylibr 137	. 2 |- (A.yE*xph -> E*x(E.y y = A /\ E.y(y = A /\ ph)))
10		simpl 102	. . . . 5 |- ((y = A /\ ph) -> y = A)
11	10	eximi 1488	. . . 4 |- (E.y(y = A /\ ph) -> E.y y = A)
12	11	ancri 307	. . 3 |- (E.y(y = A /\ ph) -> (E.y y = A /\ E.y(y = A /\ ph)))
13	12	moimi 1962	. 2 |- (E*x(E.y y = A /\ E.y(y = A /\ ph)) -> E*xE.y(y = A /\ ph))
14	9, 13	syl 14	1 |- (A.yE*xph -> E*xE.y(y = A /\ ph))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898   _Vcvv 2551

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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  mosub  2713