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  6324  fo2ndresm  6325  eloprabi  6361  fo2ndf  6392  xpsnen  7005  xpassen  7014  ac6sfi  7087  undifdc  7116  ine0  8573  nn0n0n1ge2  9550  fzval2  10246  fseq1p1m1  10329  fsum2dlemstep  12000  modfsummodlemstep  12023  fprod2dlemstep  12188  ef4p  12260  sin01bnd  12323  odd2np1  12439  sqpweven  12752  2sqpwodd  12753  psmetdmdm  15054  xmetdmdm  15086  dveflem  15456  reeff1oleme  15502  abssinper  15576  lgseisenlem1  15805