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  3853  elxp4  5215  elxp5  5216  fo1stresm  6305  fo2ndresm  6306  eloprabi  6340  fo2ndf  6371  xpsnen  6976  xpassen  6985  ac6sfi  7056  undifdc  7082  ine0  8536  nn0n0n1ge2  9513  fzval2  10203  fseq1p1m1  10286  fsum2dlemstep  11940  modfsummodlemstep  11963  fprod2dlemstep  12128  ef4p  12200  sin01bnd  12263  odd2np1  12379  sqpweven  12692  2sqpwodd  12693  psmetdmdm  14992  xmetdmdm  15024  dveflem  15394  reeff1oleme  15440  abssinper  15514  lgseisenlem1  15743