Theorem sbeqal1i 39440

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 39439	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)
2		sbeqal1i.1	. 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)
3	1, 2	mpg 1898	1 ⊢ 𝑦 = 𝑧

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-12 2222  ax-13 2391

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070

This theorem is referenced by:  sbeqal2i  39441