Theorem 3eqtr3a 2197

Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Hypotheses

Ref	Expression
3eqtr3a.1	⊢ 𝐴 = 𝐵
3eqtr3a.2	⊢ (𝜑 → 𝐴 = 𝐶)
3eqtr3a.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3eqtr3a	⊢ (𝜑 → 𝐶 = 𝐷)

Proof of Theorem 3eqtr3a

Step	Hyp	Ref	Expression
1		3eqtr3a.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		3eqtr3a.1	. . 3 ⊢ 𝐴 = 𝐵
3		3eqtr3a.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	syl5eq 2185	. 2 ⊢ (𝜑 → 𝐴 = 𝐷)
5	1, 4	eqtr3d 2175	1 ⊢ (𝜑 → 𝐶 = 𝐷)

Syntax hints:   → wi 4   = wceq 1332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133

This theorem is referenced by:  uneqin  3332  coi2  5063  foima  5358  f1imacnv  5392  fvsnun2  5626  fnsnsplitdc  6409  phplem4  6757  phplem4on  6769  halfnqq  7242  resqrexlemcalc1  10818  absefib  11513  efieq1re  11514  restopnb  12389  cnmpt2t  12501  reeflog  12992  rpcxpsqrt  13050