Theorem sbeqal1 38598

Description: If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧 is true. (Contributed by Andrew Salmon, 2-Jun-2011.)

Assertion

Ref	Expression
sbeqal1	⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)

Distinct variable group:   𝑥,𝑧

Proof of Theorem sbeqal1

Step	Hyp	Ref	Expression
1		sb2 2352	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧)
2		equsb3 2432	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
3	1, 2	sylib 208	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)

Syntax hints:   → wi 4  ∀wal 1481  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbeqal1i  38599  sbeqalbi  38601