Theorem eleqtrrid 2321

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 2237	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	eleqtrid 2320	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  rabsnt  3750  exmid1stab  4304  0elnn  4723  canth  5979  tfrexlem  6543  rdgtfr  6583  rdgruledefgg  6584  exmidonfinlem  7447  hashinfom  11086  swrds1  11298  ennnfonelemhom  13099  fnpr2ob  13486  upgrex  16027  upgr1een  16048  wlkl1loop  16282