Theorem equvin 1652

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109.

Assertion

Ref	Expression
equvin	|- (x = y <-> E.z(x = z /\ z = y))

Distinct variable groups:   x,z   y,z

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equvini 1531	. 2 |- (x = y -> E.z(x = z /\ z = y))
2		ax-17 1317	. . 3 |- (x = y -> A.z x = y)
3		equtr 1490	. . . 4 |- (x = z -> (z = y -> x = y))
4	3	imp 377	. . 3 |- ((x = z /\ z = y) -> x = y)
5	2, 4	19.23ai 1412	. 2 |- (E.z(x = z /\ z = y) -> x = y)
6	1, 5	impbii 174	1 |- (x = y <-> E.z(x = z /\ z = y))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  E.wex 1326

This theorem is referenced by:  eqvincOLD 2388

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-17 1317  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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