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  4338  dcextest  4672  relop  4871  funopg  5351  funcnvres  5393  mapsnconst  6839  snexxph  7113  apreap  8730  recextlem1  8794  nn0supp  9417  intqfrac2  10536  hashprg  11025  hashfacen  11053  ccatrid  11137  explecnv  12011  grp1inv  13635  rnrhmsubrg  14210  rerestcntop  15226  rerest  15228  mpomulcn  15234  binom4  15647