Theorem eqtr2id 2284

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 2283	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2244	1 ⊢ (𝜑 → 𝐶 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqtr3di  2286  opeqsn  4393  dcextest  4728  relop  4930  funopg  5411  funcnvres  5454  mapsnconst  6976  snexxph  7267  apreap  8917  recextlem1  8981  nn0supp  9623  intqfrac2  10769  hashprg  11263  hashfacen  11298  ccatrid  11389  explecnv  12288  grp1inv  13961  rnrhmsubrg  14609  rerestcntop  15708  rerest  15710  mpomulcn  15716  binom4  16138  wlkvtxedg  16702  wlkres  16718