Theorem syl5eleq 2707

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 2703	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  syl5eleqr  2708  opth1  4944  opth  4945  eqelsuc  5806  tfrlem11  7484  oalimcl  7640  omlimcl  7658  frgp0  18173  txdis  21435  ordtconnlem1  29970  rankeq1o  32278