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  3857  elxp4  5222  elxp5  5223  fo1stresm  6319  fo2ndresm  6320  eloprabi  6356  fo2ndf  6387  xpsnen  7000  xpassen  7009  ac6sfi  7080  undifdc  7109  ine0  8563  nn0n0n1ge2  9540  fzval2  10236  fseq1p1m1  10319  fsum2dlemstep  11985  modfsummodlemstep  12008  fprod2dlemstep  12173  ef4p  12245  sin01bnd  12308  odd2np1  12424  sqpweven  12737  2sqpwodd  12738  psmetdmdm  15038  xmetdmdm  15070  dveflem  15440  reeff1oleme  15486  abssinper  15560  lgseisenlem1  15789