Theorem eqtr2 2194

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eqtr2	⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)

Proof of Theorem eqtr2

Step	Hyp	Ref	Expression
1		eqcom 2177	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		eqtr 2193	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
3	1, 2	sylanb 284	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)

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 1445  ax-gen 1447  ax-4 1508  ax-17 1524  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-cleq 2168

This theorem is referenced by:  eqvinc  2858  eqvincg  2859  moop2  4245  reusv3i  4453  relop  4770  f0rn0  5402  fliftfun  5787  th3qlem1  6627  enq0ref  7407  enq0tr  7408  genpdisj  7497  addlsub  8301  fsum2dlemstep  11408  0dvds  11784  cncongr1  12068