Theorem eqtr3 2056

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.)

Assertion

Ref	Expression
eqtr3	⊢ ((A = 𝐶 ∧ B = 𝐶) → A = B)

Proof of Theorem eqtr3

Step	Hyp	Ref	Expression
1		eqcom 2039	. 2 ⊢ (B = 𝐶 ↔ 𝐶 = B)
2		eqtr 2054	. 2 ⊢ ((A = 𝐶 ∧ 𝐶 = B) → A = B)
3	1, 2	sylan2b 271	1 ⊢ ((A = 𝐶 ∧ B = 𝐶) → A = B)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030

This theorem is referenced by:  eueq  2706  euind  2722  reuind  2738  preqsn  3537  eusv1  4150  funopg  4877  foco  5059  mpt2fun  5545  enq0tr  6417  receuap  7432  xrltso  8487  xrlttri3  8488