Theorem equsb3lem 1824

Description: Lemma for equsb3 1825. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
equsb3lem	|- ( [ x / y ] y = z <-> x = z )

Distinct variable groups:   y, z   x, y

Proof of Theorem equsb3lem

Step	Hyp	Ref	Expression
1		ax-17 1419	. 2 |- ( x = z -> A. y x = z )
2		equequ1 1598	. 2 |- ( y = x -> ( y = z <-> x = z ) )
3	1, 2	sbieh 1673	1 |- ( [ x / y ] y = z <-> x = z )

Syntax hints:   <-> wb 98   = wceq 1243   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  equsb3  1825