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  4343  dcextest  4677  relop  4878  funopg  5358  funcnvres  5400  mapsnconst  6858  snexxph  7140  apreap  8757  recextlem1  8821  nn0supp  9444  intqfrac2  10571  hashprg  11062  hashfacen  11090  ccatrid  11174  explecnv  12056  grp1inv  13680  rnrhmsubrg  14256  rerestcntop  15272  rerest  15274  mpomulcn  15280  binom4  15693  wlkvtxedg  16160  wlkres  16174