Theorem eqtr2di 2281

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 2280	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2237	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1397

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  eqtr4id  2283  elpr2elpr  3859  elxp4  5224  elxp5  5225  fo1stresm  6323  fo2ndresm  6324  eloprabi  6360  fo2ndf  6391  xpsnen  7004  xpassen  7013  ac6sfi  7086  undifdc  7115  ine0  8572  nn0n0n1ge2  9549  fzval2  10245  fseq1p1m1  10328  fsum2dlemstep  11994  modfsummodlemstep  12017  fprod2dlemstep  12182  ef4p  12254  sin01bnd  12317  odd2np1  12433  sqpweven  12746  2sqpwodd  12747  psmetdmdm  15047  xmetdmdm  15079  dveflem  15449  reeff1oleme  15495  abssinper  15569  lgseisenlem1  15798