Theorem eqeltrrdi 2321

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

Hypotheses

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

Assertion

Ref	Expression
eqeltrrdi	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem eqeltrrdi

Step	Hyp	Ref	Expression
1		eqeltrrdi.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2235	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqeltrrdi.2	. 2 ⊢ 𝐵 ∈ 𝐶
4	2, 3	eqeltrdi 2320	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  eusvnfb  4544  releldm2  6329  mapprc  6797  ixpprc  6864  ixpssmap2g  6872  ixpssmapg  6873  bren  6893  brdomg  6895  mapen  7003  ssenen  7008  fi0  7138  nnnninf2  7290  ioof  10163  hashfacen  11053  fsum3  11893  psrval  14624  cnrehmeocntop  15278