Theorem eqtr3 2197

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 2179	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
2		eqtr 2195	. 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 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  eueq  2909  euind  2925  reuind  2943  preqsn  3776  eusv1  4453  funopg  5251  funinsn  5266  foco  5449  mpofun  5977  enq0tr  7433  lteupri  7616  elrealeu  7828  rereceu  7888  receuap  8626  xrltso  9796  xrlttri3  9797  iseqf1olemab  10489  fsumparts  11478  odd2np1  11878  grpinveu  12911  exmidsbthrlem  14773