Theorem eleqtrid 2259

Description: B 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 9	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		eleqtrid.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	eleqtrd 2249	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by:  eleqtrrid  2260  opth1  4221  opth  4222  eqelsuc  4404  txdis  13071  bj-nnelirr  13988