Theorem eqsb1lem 2296

Description: Lemma for eqsb1 2297. (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 1539	. 2 ⊢ Ⅎ𝑥 𝑦 = 𝐴
2		eqeq1 2200	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
3	1, 2	sbie 1802	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1364  [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2186

This theorem is referenced by:  eqsb1  2297