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Theorem 3eqtr3rd 2097
Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
Hypotheses
Ref Expression
3eqtr3d.1 (𝜑𝐴 = 𝐵)
3eqtr3d.2 (𝜑𝐴 = 𝐶)
3eqtr3d.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
3eqtr3rd (𝜑𝐷 = 𝐶)

Proof of Theorem 3eqtr3rd
StepHypRef Expression
1 3eqtr3d.3 . 2 (𝜑𝐵 = 𝐷)
2 3eqtr3d.1 . . 3 (𝜑𝐴 = 𝐵)
3 3eqtr3d.2 . . 3 (𝜑𝐴 = 𝐶)
42, 3eqtr3d 2090 . 2 (𝜑𝐵 = 𝐶)
51, 4eqtr3d 2090 1 (𝜑𝐷 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  fcofo  5452  fcof1o  5457  nnaword  6115  pn0sr  6914  negeu  7265  add20  7543  2halves  8211  bcnn  9625  bcpasc  9634  resqrexlemover  9837
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