Theorem eleqtrid 2862

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 2858	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794  df-cleq 2748  df-clel 2831

This theorem is referenced by:  eleqtrrid  2863  opth1  5437  opth  5438  eqelsuc  6421  tfrlem11  8347  oalimcl  8517  omlimcl  8535  frgp0  19776  txdis  23665  ordtconnlem1  34175  rankeq1o  36469  preel  38947