Theorem eleqtrrid 2230

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 2146	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	eleqtrid 2229	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  rabsnt  3606  0elnn  4540  tfrexlem  6239  rdgtfr  6279  rdgruledefgg  6280  exmidonfinlem  7066  hashinfom  10556  ennnfonelemhom  11964  exmid1stab  13368