Theorem dveeq2 1214

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-17 973	. 2 |- (z = w -> A.x z = w)
2		ax-17 973	. 2 |- (z = y -> A.w z = y)
3		equequ2 1137	. 2 |- (w = y -> (z = w <-> z = y))
4	1, 2, 3	dvelimfALT 1155	1 |- (-. A.x x = y -> (z = y -> A.x z = y))

Syntax hints:  -. wn 2   -> wi 3  A.wal 956   = wceq 958

This theorem is referenced by:  ax11v2 1217  ax11eq 1365  ax11el 1366  ax11inda 1373  nd5 4954  axrepndlem1 4956  axpowndlem2 4962  axpowndlem3 4963  axacndlem5 4975

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142

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

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