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 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 2213

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

This theorem is referenced by:  eqtr4id  2283  elpr2elpr  3864  elxp4  5231  elxp5  5232  fo1stresm  6333  fo2ndresm  6334  eloprabi  6370  fo2ndf  6401  xpsnen  7048  xpassen  7057  ac6sfi  7130  undifdc  7159  ine0  8615  nn0n0n1ge2  9594  fzval2  10291  fseq1p1m1  10374  fsum2dlemstep  12058  modfsummodlemstep  12081  fprod2dlemstep  12246  ef4p  12318  sin01bnd  12381  odd2np1  12497  sqpweven  12810  2sqpwodd  12811  psmetdmdm  15118  xmetdmdm  15150  dveflem  15520  reeff1oleme  15566  abssinper  15640  lgseisenlem1  15872