Theorem syl5eleq 2176

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 2166	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1289   ∈ wcel 1438

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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084

This theorem is referenced by:  syl5eleqr  2177  opth1  4063  opth  4064  eqelsuc  4246  bj-nnelirr  11848