Theorem 3eltr4d 2259

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 2255	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
5	1, 4	eqeltrd 2252	1 ⊢ (𝜑 → 𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2146

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 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-cleq 2168  df-clel 2171

This theorem is referenced by:  ovmpodxf  5990  nnaordi  6499  iccf1o  9973  nnmindc  12000  ennnfonelemrn  12385  ctiunctlemfo  12405  mndpropd  12705  mulgnndir  12870  srgcl  12946  srgidcl  12952  ringidcl  12996  ringpropd  13009