Theorem eqtr3 2224

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 2206	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
2		eqtr 2222	. 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 1372

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 1469  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197

This theorem is referenced by:  eueq  2943  euind  2959  reuind  2977  preqsn  3815  eusv1  4498  funopg  5304  funinsn  5322  foco  5508  funopdmsn  5763  mpofun  6046  enq0tr  7546  lteupri  7729  elrealeu  7941  rereceu  8001  receuap  8741  xrltso  9917  xrlttri3  9918  iseqf1olemab  10645  fsumparts  11723  odd2np1  12126  grpinveu  13312  exmidsbthrlem  15894