Theorem eleqtrrid 2283

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

Hypotheses

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

Assertion

Ref	Expression
eleqtrrid	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem eleqtrrid

Step	Hyp	Ref	Expression
1		eleqtrrid.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		eleqtrrid.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2199	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	eleqtrid 2282	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  rabsnt  3694  exmid1stab  4238  0elnn  4652  canth  5872  tfrexlem  6389  rdgtfr  6429  rdgruledefgg  6430  exmidonfinlem  7255  hashinfom  10852  ennnfonelemhom  12575  fnpr2ob  12926