Theorem eqtr3 2254

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 2236	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
2		eqtr 2252	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐶 = 𝐵) → 𝐴 = 𝐵)
3	1, 2	sylan2b 287	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  eueq  2991  euind  3007  reuind  3025  ssprsseq  3861  preqsn  3884  eusv1  4578  funopg  5391  funinsn  5410  foco  5606  funopdmsn  5869  mpofun  6163  enq0tr  7765  lteupri  7948  elrealeu  8160  rereceu  8220  receuap  8960  xrltso  10148  xrlttri3  10149  iseqf1olemab  10888  fsumparts  12181  odd2np1  12584  grpinveu  13793  exmidsbthrlem  16928