Theorem eleqtrrd 2117

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Hypotheses

Ref	Expression
eleqtrrd.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
eleqtrrd.2	⊢ (𝜑 → 𝐶 = 𝐵)

Assertion

Ref	Expression
eleqtrrd	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem eleqtrrd

Step	Hyp	Ref	Expression
1		eleqtrrd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		eleqtrrd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2045	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	eleqtrd 2116	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  3eltr4d  2121  tfrexlem  5948  erref  6126  en1uniel  6284  fin0  6342  fin0or  6343  prarloclemarch2  6517  fzopth  8924  fzoss2  9028  fzosubel3  9052  elfzomin  9062  elfzonlteqm1  9066  fzoend  9078  fzofzp1  9083  fzofzp1b  9084  peano2fzor  9088  frecuzrdgcl  9199  frecuzrdg0  9200  frecuzrdgsuc  9201