Theorem eqtr2id 2223

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 2222	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2183	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  eqtr3di  2225  opeqsn  4254  dcextest  4582  relop  4779  funopg  5252  funcnvres  5291  mapsnconst  6696  snexxph  6951  apreap  8546  recextlem1  8610  nn0supp  9230  intqfrac2  10321  hashprg  10790  hashfacen  10818  explecnv  11515  grp1inv  12982  rerestcntop  14135  binom4  14482