Theorem equsb3lem 1978

Description: Lemma for equsb3 1979. (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 1549	. 2 |- ( y = z -> A. x y = z )
2		equequ1 1735	. 2 |- ( x = y -> ( x = z <-> y = z ) )
3	1, 2	sbieh 1813	1 |- ( [ y / x ] x = z <-> y = z )

Syntax hints:   <-> wb 105   = wceq 1373   [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  equsb3  1979