Theorem syl5eleq 2126

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 2116	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  syl5eleqr  2127  opth1  3973  opth  3974  eqelsuc  4156  bj-nnelirr  10078