Theorem ax17eq 1253

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1012 considered as a metatheorem. Do not use it for later proofs - use ax-17 1012 instead, to avoid reference to the redundant axiom ax-16 1252.)

Assertion

Ref	Expression
ax17eq	|- (x = y -> A.z x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem ax17eq

Step	Hyp	Ref	Expression
1		ax-12 1009	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
2		ax-16 1252	. 2 |- (A.z z = x -> (x = y -> A.z x = y))
3		ax-16 1252	. 2 |- (A.z z = y -> (x = y -> A.z x = y))
4	1, 2, 3	pm2.61ii 136	1 |- (x = y -> A.z x = y)

Syntax hints:   -> wi 3  A.wal 995   = wceq 997

This theorem is referenced by:  dveeq2ALT 1255  dveeq1ALT 1397  euequ1OLD 1501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-12 1009  ax-16 1252

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