Theorem a12lem2 1375

Description: Proof of second hypothesis of a12study 1376.

Assertion

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

Proof of Theorem a12lem2

Step	Hyp	Ref	Expression
1		equcom 1127	. . . . . 6 |- (z = x <-> x = z)
2	1	imbi1i 186	. . . . 5 |- ((z = x -> -. z = y) <-> (x = z -> -. z = y))
3		imnan 242	. . . . 5 |- ((x = z -> -. z = y) <-> -. (x = z /\ z = y))
4	2, 3	bitr 173	. . . 4 |- ((z = x -> -. z = y) <-> -. (x = z /\ z = y))
5	4	albii 997	. . 3 |- (A.z(z = x -> -. z = y) <-> A.z -. (x = z /\ z = y))
6		alnex 1031	. . 3 |- (A.z -. (x = z /\ z = y) <-> -. E.z(x = z /\ z = y))
7	5, 6	bitr 173	. 2 |- (A.z(z = x -> -. z = y) <-> -. E.z(x = z /\ z = y))
8		equvini 1166	. . 3 |- (x = y -> E.z(x = z /\ z = y))
9	8	con3i 98	. 2 |- (-. E.z(x = z /\ z = y) -> -. x = y)
10	7, 9	sylbi 199	1 |- (A.z(z = x -> -. z = y) -> -. x = y)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 952   = wceq 954  E.wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979

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