Theorem eqtr3 2251

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 2233	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
2		eqtr 2249	. 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 1397

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  eueq  2977  euind  2993  reuind  3011  ssprsseq  3835  preqsn  3858  eusv1  4549  funopg  5360  funinsn  5379  foco  5570  funopdmsn  5833  mpofun  6122  enq0tr  7653  lteupri  7836  elrealeu  8048  rereceu  8108  receuap  8848  xrltso  10030  xrlttri3  10031  iseqf1olemab  10763  fsumparts  12030  odd2np1  12433  grpinveu  13620  exmidsbthrlem  16626