Theorem equsb3lem 2430

Description: Lemma for equsb3 2431. (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 1840	. 2 ⊢ Ⅎ𝑦 𝑥 = 𝑧
2		equequ1 1949	. 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
3	1, 2	sbie 2407	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Syntax hints:   ↔ wb 196  [wsb 1877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878

This theorem is referenced by:  equsb3  2431  equsb3ALT  2432