Theorem eqtr2id 2242

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 2241	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2202	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  eqtr3di  2244  opeqsn  4285  dcextest  4617  relop  4816  funopg  5292  funcnvres  5331  mapsnconst  6753  snexxph  7016  apreap  8614  recextlem1  8678  nn0supp  9301  intqfrac2  10411  hashprg  10900  hashfacen  10928  explecnv  11670  grp1inv  13239  rnrhmsubrg  13808  rerestcntop  14794  rerest  14796  mpomulcn  14802  binom4  15215