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  8437  nn0n0n1ge2  9413  fzval2  10103  fseq1p1m1  10186  fsum2dlemstep  11616  modfsummodlemstep  11639  fprod2dlemstep  11804  ef4p  11876  sin01bnd  11939  odd2np1  12055  sqpweven  12368  2sqpwodd  12369  psmetdmdm  14644  xmetdmdm  14676  dveflem  15046  reeff1oleme  15092  abssinper  15166  lgseisenlem1  15395