Theorem eqtr2di 2279

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 2278	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2235	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eqtr4id  2281  elpr2elpr  3854  elxp4  5216  elxp5  5217  fo1stresm  6313  fo2ndresm  6314  eloprabi  6348  fo2ndf  6379  xpsnen  6988  xpassen  6997  ac6sfi  7068  undifdc  7097  ine0  8551  nn0n0n1ge2  9528  fzval2  10219  fseq1p1m1  10302  fsum2dlemstep  11960  modfsummodlemstep  11983  fprod2dlemstep  12148  ef4p  12220  sin01bnd  12283  odd2np1  12399  sqpweven  12712  2sqpwodd  12713  psmetdmdm  15013  xmetdmdm  15045  dveflem  15415  reeff1oleme  15461  abssinper  15535  lgseisenlem1  15764