Theorem eqtr2di 2246

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 2245	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2202	1 ⊢ (𝜑 → 𝐶 = 𝐴)

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:  eqtr4id  2248  elxp4  5158  elxp5  5159  fo1stresm  6228  fo2ndresm  6229  eloprabi  6263  fo2ndf  6294  xpsnen  6889  xpassen  6898  ac6sfi  6968  undifdc  6994  ine0  8439  nn0n0n1ge2  9415  fzval2  10105  fseq1p1m1  10188  fsum2dlemstep  11618  modfsummodlemstep  11641  fprod2dlemstep  11806  ef4p  11878  sin01bnd  11941  odd2np1  12057  sqpweven  12370  2sqpwodd  12371  psmetdmdm  14668  xmetdmdm  14700  dveflem  15070  reeff1oleme  15116  abssinper  15190  lgseisenlem1  15419