Theorem 3eqtr3rd 2271

Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)

Hypotheses

Ref	Expression
3eqtr3d.1	⊢ (𝜑 → 𝐴 = 𝐵)
3eqtr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3eqtr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3eqtr3rd	⊢ (𝜑 → 𝐷 = 𝐶)

Proof of Theorem 3eqtr3rd

Step	Hyp	Ref	Expression
1		3eqtr3d.3	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
2		3eqtr3d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3		3eqtr3d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
4	2, 3	eqtr3d 2264	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
5	1, 4	eqtr3d 2264	1 ⊢ (𝜑 → 𝐷 = 𝐶)

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:  fcofo  5914  fcof1o  5919  frecabcl  6551  nnaword  6665  nninfisol  7308  enomnilem  7313  fodju0  7322  enmkvlem  7336  enwomnilem  7344  pn0sr  7966  negeu  8345  add20  8629  2halves  9348  bcnn  10987  bcpasc  10996  wrdeqs1cat  11260  resqrexlemover  11529  fsumneg  11970  geolim  12030  geolim2  12031  mertensabs  12056  sincossq  12267  demoivre  12292  eirraplem  12296  gcdid  12515  gcdmultipled  12522  phiprmpw  12752  pythagtriplem12  12806  expnprm  12884  imasbas  13348  imasplusg  13349  imasmulr  13350  grpinvid1  13593  grpnpcan  13633  grplactcnv  13643  ghmgrp  13663  conjghm  13821  ringnegl  14022  ringnegr  14023  ringmneg2  14025  ring1  14030  rdivmuldivd  14116  lmodfopne  14298  lmodvsneg  14303  ioo2bl  15233  ptolemy  15506  coskpi  15530  logbgcd1irr  15649  logbgcd1irraplemap  15651  lgseisenlem3  15759  lgseisenlem4  15760  lgsquadlem1  15764  lgsquadlem2  15765