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  4497  funopg  5302  funinsn  5317  foco  5503  mpofun  6037  enq0tr  7529  lteupri  7712  elrealeu  7924  rereceu  7984  receuap  8724  xrltso  9900  xrlttri3  9901  iseqf1olemab  10628  fsumparts  11700  odd2np1  12103  grpinveu  13288  exmidsbthrlem  15825