Theorem eqeltrrdi 2323

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 2237	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqeltrrdi.2	. 2 ⊢ 𝐵 ∈ 𝐶
4	2, 3	eqeltrdi 2322	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  eusvnfb  4551  releldm2  6348  mapprc  6821  ixpprc  6888  ixpssmap2g  6896  ixpssmapg  6897  bren  6917  brdomg  6919  mapen  7032  ssenen  7037  fi0  7174  nnnninf2  7326  ioof  10206  hashfacen  11101  fsum3  11953  psrval  14686  cnrehmeocntop  15340