Theorem 3eqtr3a 2137

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 2125	. 2 ⊢ (𝜑 → 𝐴 = 𝐷)
5	1, 4	eqtr3d 2115	1 ⊢ (𝜑 → 𝐶 = 𝐷)

Syntax hints:   → wi 4   = wceq 1284

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  uneqin  3215  coi2  4857  foima  5131  f1imacnv  5163  fvsnun2  5382  phplem4  6341  phplem4on  6353  halfnqq  6600  resqrexlemcalc1  9900