Theorem equsb3lem 2564

Description: Lemma for equsb3 2565. (Contributed by Raph Levien and FL, 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		nfv 1988	. 2 ⊢ Ⅎ𝑦 𝑥 = 𝑧
2		equequ1 2103	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
3	1, 2	sbie 2541	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Syntax hints:   ↔ wb 196  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-10 2164  ax-12 2192  ax-13 2387

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1850  df-nf 1855  df-sb 2043

This theorem is referenced by:  equsb3  2565  equsb3ALT  2566