Theorem syl5eleq 2694

Description: A 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 11	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		syl5eleq.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	eleqtrd 2690	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606

This theorem is referenced by:  syl5eleqr  2695  opth1  4870  opth  4871  eqelsuc  5723  tfrlem11  7371  oalimcl  7527  omlimcl  7545  frgp0  17996  txdis  21245  ordtconlem1  29298  rankeq1o  31448