Theorem abeq2i 2349

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 2305	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2226	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  rabid  2727  vex  2824  csbco  3157  csbcow  3158  csbnestgf  3200  ifmdc  3680  pwss  3704  snsspw  3884  iunpw  4621  ordon  4628  funcnv3  5438  tfrlem4  6574  tfrlem8  6579  tfrlem9  6580  tfrlemibxssdm  6588  tfr1onlembxssdm  6604  tfrcllembxssdm  6617  ixpm  7002  mapsnen  7090  sbthlem1  7264  1idprl  7947  1idpru  7948  recexprlem1ssl  7990  recexprlem1ssu  7991  recexprlemss1l  7992  recexprlemss1u  7993  txbas  15282