Theorem abeq2i 2340

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 2296	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2217	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by:  rabid  2707  vex  2802  csbco  3134  csbcow  3135  csbnestgf  3177  ifmdc  3645  pwss  3665  snsspw  3841  iunpw  4570  ordon  4577  funcnv3  5382  tfrlem4  6457  tfrlem8  6462  tfrlem9  6463  tfrlemibxssdm  6471  tfr1onlembxssdm  6487  tfrcllembxssdm  6500  ixpm  6875  mapsnen  6962  sbthlem1  7120  1idprl  7773  1idpru  7774  recexprlem1ssl  7816  recexprlem1ssu  7817  recexprlemss1l  7818  recexprlemss1u  7819  txbas  14926