Theorem eqtr3 2249

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 2231	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
2		eqtr 2247	. 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 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eueq  2974  euind  2990  reuind  3008  ssprsseq  3830  preqsn  3853  eusv1  4543  funopg  5352  funinsn  5370  foco  5561  funopdmsn  5823  mpofun  6112  enq0tr  7632  lteupri  7815  elrealeu  8027  rereceu  8087  receuap  8827  xrltso  10004  xrlttri3  10005  iseqf1olemab  10736  fsumparts  11996  odd2np1  12399  grpinveu  13586  exmidsbthrlem  16450