Theorem eqtr2di 2282

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 2281	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2238	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225

This theorem is referenced by:  eqtr4id  2284  elpr2elpr  3880  elxp4  5250  elxp5  5251  fo1stresm  6355  fo2ndresm  6356  eloprabi  6392  fo2ndf  6423  xpsnen  7072  xpassen  7081  ac6sfi  7155  undifdc  7184  ine0  8667  nn0n0n1ge2  9648  fzval2  10345  fseq1p1m1  10428  hashfibclem  11206  fsum2dlemstep  12120  modfsummodlemstep  12143  fprod2dlemstep  12308  ef4p  12380  sin01bnd  12443  odd2np1  12559  sqpweven  12872  2sqpwodd  12873  psmetdmdm  15189  xmetdmdm  15221  dveflem  15591  reeff1oleme  15637  abssinper  15711  lgseisenlem1  15943