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  3529  dfopg  3806  xpid11  4889  sqxpeq0  5093  cores2  5182  funcoeqres  5535  dftpos2  6319  ine0  8420  fisumcom2  11603  fisum0diag2  11612  mertenslemi1  11700  fprodcom2fi  11791  fprodmodd  11806  4sqlem10  12556  setsslnid  12730  xpsff1o  12992  eqglact  13355  oppr1g  13638  dvmptccn  14951  dvmptc  14953  dvmptfsum  14961  fsumdvdsmul  15227  nninffeq  15664