Theorem sbeqal1 44417

Description: If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (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 2484	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → [𝑦 / 𝑥]𝑥 = 𝑧)
2		equsb3 2103	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
3	1, 2	sylib 218	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)

Syntax hints:   → wi 4  ∀wal 1538  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by:  sbeqal1i  44418