Theorem a12lem2 1419

Description: Proof of second hypothesis of a12study 1420.

Assertion

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

Proof of Theorem a12lem2

Step	Hyp	Ref	Expression
1		equcom 1171	. . . . . 6 |- (z = x <-> x = z)
2	1	imbi1i 193	. . . . 5 |- ((z = x -> -. z = y) <-> (x = z -> -. z = y))
3		imnan 249	. . . . 5 |- ((x = z -> -. z = y) <-> -. (x = z /\ z = y))
4	2, 3	bitri 180	. . . 4 |- ((z = x -> -. z = y) <-> -. (x = z /\ z = y))
5	4	albii 1040	. . 3 |- (A.z(z = x -> -. z = y) <-> A.z -. (x = z /\ z = y))
6		alnex 1074	. . 3 |- (A.z -. (x = z /\ z = y) <-> -. E.z(x = z /\ z = y))
7	5, 6	bitri 180	. 2 |- (A.z(z = x -> -. z = y) <-> -. E.z(x = z /\ z = y))
8		equvini 1210	. . 3 |- (x = y -> E.z(x = z /\ z = y))
9	8	con3i 104	. 2 |- (-. E.z(x = z /\ z = y) -> -. x = y)
10	7, 9	sylbi 206	1 |- (A.z(z = x -> -. z = y) -> -. x = y)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 230  A.wal 995   = wceq 997  E.wex 1021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-12 1009  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022

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