Theorem eqsb1lem 2453

Description: Lemma for eqsb1 2454. (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 1619	. 2 ⊢ Ⅎx y = A
2		eqeq1 2359	. 2 ⊢ (x = y → (x = A ↔ y = A))
3	1, 2	sbie 2038	1 ⊢ ([y / x]x = A ↔ y = A)

Syntax hints:   ↔ wb 176   = wceq 1642  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346

This theorem is referenced by:  eqsb1  2454