Theorem eleqtrid 2850

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

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

Assertion

Ref	Expression
eleqtrid	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem eleqtrid

Step	Hyp	Ref	Expression
1		eleqtrid.1	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		eleqtrid.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	eleqtrd 2846	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-clel 2819

This theorem is referenced by:  eleqtrrid  2851  opth1  5495  opth  5496  eqelsuc  6481  tfrlem11  8446  oalimcl  8618  omlimcl  8636  frgp0  19804  txdis  23663  ordtconnlem1  33872  rankeq1o  36137