Theorem dveeq2 1210

Description: Quantifier introduction when one pair of variables is distinct.

Assertion

Ref	Expression
dveeq2	|- (-. A.x x = y -> (z = y -> A.x z = y))

Distinct variable group:   x,z

Proof of Theorem dveeq2

Step	Hyp	Ref	Expression
1		ax-17 969	. 2 |- (z = w -> A.x z = w)
2		ax-17 969	. 2 |- (z = y -> A.w z = y)
3		equequ2 1133	. 2 |- (w = y -> (z = w <-> z = y))
4	1, 2, 3	dvelimfALT 1151	1 |- (-. A.x x = y -> (z = y -> A.x z = y))

Syntax hints:  -. wn 2   -> wi 3  A.wal 952   = wceq 954

This theorem is referenced by:  ax11v2 1213  ax11eq 1361  ax11el 1362  ax11inda 1369  nd5 4922  axrepndlem1 4924  axpowndlem2 4930  axpowndlem3 4931  axacndlem5 4943

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-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138

This theorem depends on definitions:  df-bi 147  df-an 225

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