Theorem eqsb1lem 2308

Description: Lemma for eqsb1 2309. (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 1551	. 2 |- F/ x y = A
2		eqeq1 2212	. 2 |- ( x = y -> ( x = A <-> y = A ) )
3	1, 2	sbie 1814	1 |- ( [ y / x ] x = A <-> y = A )

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-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198

This theorem is referenced by:  eqsb1  2309