Theorem 3eqtr3g 2261

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 2252	. 2 ⊢ (𝜑 → 𝐶 = 𝐵)
4		3eqtr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtrdi 2254	1 ⊢ (𝜑 → 𝐶 = 𝐷)

Syntax hints:   → wi 4   = wceq 1373

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  csbnest1g  3149  disjdif2  3539  dfopg  3817  xpid11  4902  sqxpeq0  5107  cores2  5196  funcoeqres  5555  dftpos2  6349  ine0  8468  fisumcom2  11782  fisum0diag2  11791  mertenslemi1  11879  fprodcom2fi  11970  fprodmodd  11985  bitsinv1  12306  4sqlem10  12743  setsslnid  12917  xpsff1o  13214  eqglact  13594  oppr1g  13877  dvmptccn  15220  dvmptc  15222  dvmptfsum  15230  fsumdvdsmul  15496  nninffeq  15994