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  4286  dcextest  4618  relop  4817  funopg  5293  funcnvres  5332  mapsnconst  6762  snexxph  7025  apreap  8631  recextlem1  8695  nn0supp  9318  intqfrac2  10428  hashprg  10917  hashfacen  10945  explecnv  11687  grp1inv  13309  rnrhmsubrg  13884  rerestcntop  14878  rerest  14880  mpomulcn  14886  binom4  15299