Theorem sbeqal2i 45211

Description: If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.)

Hypothesis

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

Assertion

Ref	Expression
sbeqal2i	⊢ 𝑧 = 𝑦

Distinct variable group:   𝑥,𝑧

Proof of Theorem sbeqal2i

Step	Hyp	Ref	Expression
1		sbeqal1i.1	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)
2	1	sbeqal1i 45210	. 2 ⊢ 𝑦 = 𝑧
3	2	eqcomi 2771	1 ⊢ 𝑧 = 𝑦

This proof depends on syntax axioms:

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-9 2155  ax-10 2178  ax-12 2215  ax-13 2403  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-cleq 2754

This theorem is used by: (None)