Theorem cleqtr 1118

Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13.

Assertion

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

Proof of Theorem cleqtr

Step	Hyp	Ref	Expression
1		cleq1 1107	. 2 ⊢ (A = B → (A = C ↔ B = C))
2	1	biimpar 325	1 ⊢ ((A = B ∧ B = C) → A = C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091

This theorem is referenced by:  moop2 1910  oawordeulem 3156  ider 3208  xpmapenlem4 3394  inf5 3472  aceq5lem4 3561  cfom 3710  uzind 4603

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097

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