Theorem abeq2i 2304

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 2260	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2181	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189

This theorem is referenced by:  rabid  2670  vex  2763  csbco  3091  csbcow  3092  csbnestgf  3134  ifmdc  3598  pwss  3618  snsspw  3791  iunpw  4512  ordon  4519  funcnv3  5317  tfrlem4  6368  tfrlem8  6373  tfrlem9  6374  tfrlemibxssdm  6382  tfr1onlembxssdm  6398  tfrcllembxssdm  6411  ixpm  6786  mapsnen  6867  sbthlem1  7018  1idprl  7652  1idpru  7653  recexprlem1ssl  7695  recexprlem1ssu  7696  recexprlemss1l  7697  recexprlemss1u  7698  txbas  14437