Theorem eqtr 2211

Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)

Assertion

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

Proof of Theorem eqtr

Step	Hyp	Ref	Expression
1		eqeq1 2200	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	biimpar 297	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  eqtr2  2212  eqtr3  2213  sylan9eq  2246  eqvinc  2884  eqvincg  2885  uneqdifeqim  3533  preqsn  3802  dtruex  4592  relresfld  5196  relcoi1  5198  eqer  6621  xpider  6662  addlsub  8391  bj-findis  15541