Theorem eleqtrid 2845

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 2841	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816

This theorem is referenced by:  eleqtrrid  2846  opth1  5390  opth  5391  eqelsuc  6342  tfrlem11  8208  oalimcl  8380  omlimcl  8398  frgp0  19355  txdis  22772  ordtconnlem1  31861  rankeq1o  34460