Theorem eqtr2id 2275

Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Hypotheses

Ref	Expression
eqtr2id.1	⊢ 𝐴 = 𝐵
eqtr2id.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
eqtr2id	⊢ (𝜑 → 𝐶 = 𝐴)

Proof of Theorem eqtr2id

Step	Hyp	Ref	Expression
1		eqtr2id.1	. . 3 ⊢ 𝐴 = 𝐵
2		eqtr2id.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
3	1, 2	eqtrid 2274	. 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:  eqtr3di  2277  opeqsn  4339  dcextest  4673  relop  4872  funopg  5352  funcnvres  5394  mapsnconst  6849  snexxph  7128  apreap  8745  recextlem1  8809  nn0supp  9432  intqfrac2  10553  hashprg  11043  hashfacen  11071  ccatrid  11155  explecnv  12031  grp1inv  13655  rnrhmsubrg  14231  rerestcntop  15247  rerest  15249  mpomulcn  15255  binom4  15668  wlkvtxedg  16104  wlkres  16118