Theorem syl5eleq 2167

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

Hypotheses

Ref	Expression
syl5eleq.1	⊢ 𝐴 ∈ 𝐵
syl5eleq.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
syl5eleq	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem syl5eleq

Step	Hyp	Ref	Expression
1		syl5eleq.1	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		syl5eleq.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	eleqtrd 2157	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  syl5eleqr  2168  opth1  3991  opth  3992  eqelsuc  4174  bj-nnelirr  10748