Theorem a12lem2 1768

Description: Proof of second hypothesis of a12study 1769.

Assertion

Ref	Expression
a12lem2	|- (A.z(z = x -> -. z = y) -> -. x = y)

Proof of Theorem a12lem2

Step	Hyp	Ref	Expression
1		equcom 1488	. . . . . 6 |- (z = x <-> x = z)
2	1	imbi1i 203	. . . . 5 |- ((z = x -> -. z = y) <-> (x = z -> -. z = y))
3		imnan 261	. . . . 5 |- ((x = z -> -. z = y) <-> -. (x = z /\ z = y))
4	2, 3	bitri 190	. . . 4 |- ((z = x -> -. z = y) <-> -. (x = z /\ z = y))
5	4	albii 1346	. . 3 |- (A.z(z = x -> -. z = y) <-> A.z -. (x = z /\ z = y))
6		alnex 1380	. . 3 |- (A.z -. (x = z /\ z = y) <-> -. E.z(x = z /\ z = y))
7	5, 6	bitri 190	. 2 |- (A.z(z = x -> -. z = y) <-> -. E.z(x = z /\ z = y))
8		equvini 1531	. . 3 |- (x = y -> E.z(x = z /\ z = y))
9	8	con3i 114	. 2 |- (-. E.z(x = z /\ z = y) -> -. x = y)
10	7, 9	sylbi 216	1 |- (A.z(z = x -> -. z = y) -> -. x = y)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327

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