Theorem syl6eqelr 2180

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

Hypotheses

Ref	Expression
syl6eqelr.1	⊢ (𝜑 → 𝐵 = 𝐴)
syl6eqelr.2	⊢ 𝐵 ∈ 𝐶

Assertion

Ref	Expression
syl6eqelr	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem syl6eqelr

Step	Hyp	Ref	Expression
1		syl6eqelr.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2094	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		syl6eqelr.2	. 2 ⊢ 𝐵 ∈ 𝐶
4	2, 3	syl6eqel 2179	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1290   ∈ wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085

This theorem is referenced by:  eusvnfb  4291  releldm2  5971  mapprc  6425  ixpprc  6492  ixpssmap2g  6500  ixpssmapg  6501  bren  6520  brdomg  6521  mapen  6618  ssenen  6623  ioof  9452  hashfacen  10304  fisum  10841