Theorem eqsb1lem 2332

Description: Lemma for eqsb1 2333. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
eqsb1lem	|- ( [ y / x ] x = A <-> y = A )

Distinct variable groups:   x, y   x, A

Allowed substitution hint:   A( y)

Proof of Theorem eqsb1lem

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 |- F/ x y = A
2		eqeq1 2236	. 2 |- ( x = y -> ( x = A <-> y = A ) )
3	1, 2	sbie 1837	1 |- ( [ y / x ] x = A <-> y = A )

Syntax hints:   <-> wb 105   = wceq 1395   [wsb 1808

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222

This theorem is referenced by:  eqsb1  2333