Theorem 3eltr3d 2845

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr3d.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3eltr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3eltr3d	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr3d

Step	Hyp	Ref	Expression
1		3eltr3d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		3eltr3d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		3eltr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	eleqtrd 2833	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5	1, 4	eqeltrrd 2832	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806

This theorem is referenced by:  axcc2lem  10327  axcclem  10348  icoshftf1o  13374  lincmb01cmp  13395  fzosubel  13624  symgsubmefmndALT  19316  psgnunilem1  19406  efgcpbllemb  19668  lspprabs  21030  cnmpt2res  23593  xpstopnlem1  23725  tususp  24187  tustps  24188  ressxms  24441  ressms  24442  tmsxpsval  24454  limcco  25822  dvcnp2  25849  dvcnp2OLD  25850  dvmulbr  25869  dvcobr  25877  dvcnvlem  25908  taylthlem2  26310  taylthlem2OLD  26311  jensen  26927  f1otrg  28850  nsgqusf1olem1  33376  txomap  33845  probmeasb  34441  fsum2dsub  34618  cvmlift2lem9  35353  prdsbnd2  37841  iocopn  45566  icoopn  45571  reclimc  45697  cncfiooicclem1  45937  itgiccshift  46024  dirkercncflem4  46150  fourierdlem32  46183  fourierdlem33  46184  fourierdlem60  46210  fourierdlem61  46211  fourierdlem76  46226  fourierdlem81  46231  fourierdlem90  46240  fourierdlem111  46261  uptrlem3  49250  fuco2eld3  49353  fucoid2  49387