Theorem abeq2i 2251

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)

Hypothesis

Ref	Expression
abeqi.1	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
abeq2i	⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Proof of Theorem abeq2i

Step	Hyp	Ref	Expression
1		abeqi.1	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑}
2	1	eleq2i 2207	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2128	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 183	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136

This theorem is referenced by:  rabid  2609  vex  2692  csbco  3017  csbnestgf  3057  ifmdc  3514  pwss  3531  snsspw  3699  iunpw  4409  ordon  4410  funcnv3  5193  tfrlem4  6218  tfrlem8  6223  tfrlem9  6224  tfrlemibxssdm  6232  tfr1onlembxssdm  6248  tfrcllembxssdm  6261  ixpm  6632  mapsnen  6713  sbthlem1  6853  1idprl  7422  1idpru  7423  recexprlem1ssl  7465  recexprlem1ssu  7466  recexprlemss1l  7467  recexprlemss1u  7468  txbas  12466