Theorem eqtr2di 2288

Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Hypotheses

Ref	Expression
eqtr2di.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqtr2di.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
eqtr2di	⊢ (𝜑 → 𝐶 = 𝐴)

Proof of Theorem eqtr2di

Step	Hyp	Ref	Expression
1		eqtr2di.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqtr2di.2	. . 3 ⊢ 𝐵 = 𝐶
3	1, 2	eqtrdi 2287	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2244	1 ⊢ (𝜑 → 𝐶 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqtr4id  2290  elpr2elpr  3901  elxp4  5275  elxp5  5276  fo1stresm  6395  fo2ndresm  6396  eloprabi  6432  fo2ndf  6463  xpsnen  7119  xpassen  7128  ac6sfi  7202  undifdc  7231  ine0  8721  nn0n0n1ge2  9715  fzval2  10414  fseq1p1m1  10501  hashfibclem  11282  hashf1  11287  fsum2dlemstep  12201  modfsummodlemstep  12224  fprod2dlemstep  12389  ef4p  12461  sin01bnd  12524  odd2np1  12640  sqpweven  12953  2sqpwodd  12954  psmetdmdm  15425  xmetdmdm  15457  dveflem  15827  reeff1oleme  15873  abssinper  15947  lgseisenlem1  16189