Theorem equvin 1274

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 1167	. 2 |- (x = y -> E.z(x = z /\ z = y))
2		ax-17 970	. . 3 |- (x = y -> A.z x = y)
3		equtr 1130	. . . 4 |- (x = z -> (z = y -> x = y))
4	3	imp 350	. . 3 |- ((x = z /\ z = y) -> x = y)
5	2, 4	19.23ai 1063	. 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 955  E.wex 979

This theorem is referenced by:  eqvinc 1880

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980

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