Theorem equsb3lem 1873

Description: Lemma for equsb3 1874. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
equsb3lem	⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Distinct variable groups:   𝑦,𝑧   𝑥,𝑦

Proof of Theorem equsb3lem

Step	Hyp	Ref	Expression
1		ax-17 1465	. 2 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)
2		equequ1 1646	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
3	1, 2	sbieh 1721	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Syntax hints:   ↔ wb 104   = wceq 1290  [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-sb 1694

This theorem is referenced by:  equsb3  1874