Theorem abeq2i 2248

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 2204	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2125	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 183	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133

This theorem is referenced by:  rabid  2604  vex  2684  csbco  3008  csbnestgf  3047  ifmdc  3504  pwss  3521  snsspw  3686  iunpw  4396  ordon  4397  funcnv3  5180  tfrlem4  6203  tfrlem8  6208  tfrlem9  6209  tfrlemibxssdm  6217  tfr1onlembxssdm  6233  tfrcllembxssdm  6246  ixpm  6617  mapsnen  6698  sbthlem1  6838  1idprl  7391  1idpru  7392  recexprlem1ssl  7434  recexprlem1ssu  7435  recexprlemss1l  7436  recexprlemss1u  7437  txbas  12416