Theorem eqtr2di 2284

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 2283	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2240	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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  eqtr4id  2286  elpr2elpr  3885  elxp4  5255  elxp5  5256  fo1stresm  6368  fo2ndresm  6369  eloprabi  6405  fo2ndf  6436  xpsnen  7085  xpassen  7094  ac6sfi  7168  undifdc  7197  ine0  8684  nn0n0n1ge2  9665  fzval2  10364  fseq1p1m1  10450  hashfibclem  11231  fsum2dlemstep  12145  modfsummodlemstep  12168  fprod2dlemstep  12333  ef4p  12405  sin01bnd  12468  odd2np1  12584  sqpweven  12897  2sqpwodd  12898  psmetdmdm  15315  xmetdmdm  15347  dveflem  15717  reeff1oleme  15763  abssinper  15837  lgseisenlem1  16069