Theorem eleqtrid 2266

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 2256	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  eleqtrrid  2267  opth1  4237  opth  4238  eqelsuc  4420  2omotaplemst  7257  txdis  13780  bj-nnelirr  14708