Theorem 3eltr4d 2261

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

Hypotheses

Ref	Expression
3eltr4d.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
3eltr4d.2	⊢ (𝜑 → 𝐶 = 𝐴)
3eltr4d.3	⊢ (𝜑 → 𝐷 = 𝐵)

Assertion

Ref	Expression
3eltr4d	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Proof of Theorem 3eltr4d

Step	Hyp	Ref	Expression
1		3eltr4d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐴)
2		3eltr4d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		3eltr4d.3	. . 3 ⊢ (𝜑 → 𝐷 = 𝐵)
4	2, 3	eleqtrrd 2257	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5	1, 4	eqeltrd 2254	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148

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-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  ovmpodxf  6002  nnaordi  6511  iccf1o  10006  nnmindc  12037  ennnfonelemrn  12422  ctiunctlemfo  12442  mndpropd  12846  issubmnd  12848  mulgnndir  13017  subg0cl  13047  subginvcl  13048  subgcl  13049  srgcl  13158  srgidcl  13164  ringidcl  13208  ringpropd  13222  dvdsrd  13268  dvrvald  13308  subrgmcl  13359  subrgunit  13365  lmodprop2d  13443