Theorem sbeqal1i 45210

Description: Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.)

Hypothesis

Ref	Expression
sbeqal1i.1	⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)

Assertion

Ref	Expression
sbeqal1i	⊢ 𝑦 = 𝑧

Distinct variable group:   𝑥,𝑧

Proof of Theorem sbeqal1i

Step	Hyp	Ref	Expression
1		sbeqal1 45209	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)
2		sbeqal1i.1	. 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)
3	1, 2	mpg 1830	1 ⊢ 𝑦 = 𝑧

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2215  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sbeqal2i  45211