Theorem equsb3lem 1950

Description: Lemma for equsb3 1951. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

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

Distinct variable groups:   x, z   x, y

Proof of Theorem equsb3lem

Step	Hyp	Ref	Expression
1		ax-17 1526	. 2 |- ( y = z -> A. x y = z )
2		equequ1 1712	. 2 |- ( x = y -> ( x = z <-> y = z ) )
3	1, 2	sbieh 1790	1 |- ( [ y / x ] x = z <-> y = z )

Syntax hints:   <-> wb 105   = wceq 1353   [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  equsb3  1951