Theorem equvin 1275

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 1168	. 2 |- (x = y -> E.z(x = z /\ z = y))
2		ax-17 971	. . 3 |- (x = y -> A.z x = y)
3		equtr 1131	. . . 4 |- (x = z -> (z = y -> x = y))
4	3	imp 350	. . 3 |- ((x = z /\ z = y) -> x = y)
5	2, 4	19.23ai 1064	. 2 |- (E.z(x = z /\ z = y) -> x = y)
6	1, 5	impbi 157	1 |- (x = y <-> E.z(x = z /\ z = y))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956  E.wex 980

This theorem is referenced by:  eqvinc 1883

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981

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