Theorem 3eqtr3a 2261

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	eqtrid 2249	. 2 ⊢ (𝜑 → 𝐴 = 𝐷)
5	1, 4	eqtr3d 2239	1 ⊢ (𝜑 → 𝐶 = 𝐷)

Syntax hints:   → wi 4   = wceq 1372

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 1469  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197

This theorem is referenced by:  uneqin  3423  coi2  5198  foima  5502  f1imacnv  5538  fvsnun2  5781  fnsnsplitdc  6590  phplem4  6951  phplem4on  6963  halfnqq  7522  resqrexlemcalc1  11267  absefib  12024  efieq1re  12025  restopnb  14595  cnmpt2t  14707  reeflog  15277  rpcxpsqrt  15336