Theorem 3eqtr3g 2252

Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Hypotheses

Ref	Expression
3eqtr3g.1	⊢ (𝜑 → 𝐴 = 𝐵)
3eqtr3g.2	⊢ 𝐴 = 𝐶
3eqtr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eqtr3g	⊢ (𝜑 → 𝐶 = 𝐷)

Proof of Theorem 3eqtr3g

Step	Hyp	Ref	Expression
1		3eqtr3g.2	. . 3 ⊢ 𝐴 = 𝐶
2		3eqtr3g.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	eqtr3id 2243	. 2 ⊢ (𝜑 → 𝐶 = 𝐵)
4		3eqtr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtrdi 2245	1 ⊢ (𝜑 → 𝐶 = 𝐷)

Syntax hints:   → wi 4   = 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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  csbnest1g  3140  disjdif2  3530  dfopg  3807  xpid11  4890  sqxpeq0  5094  cores2  5183  funcoeqres  5538  dftpos2  6328  ine0  8439  fisumcom2  11622  fisum0diag2  11631  mertenslemi1  11719  fprodcom2fi  11810  fprodmodd  11825  bitsinv1  12146  4sqlem10  12583  setsslnid  12757  xpsff1o  13053  eqglact  13433  oppr1g  13716  dvmptccn  15059  dvmptc  15061  dvmptfsum  15069  fsumdvdsmul  15335  nninffeq  15775