Theorem abeq2i 2315

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 2271	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2192	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1372   ∈ wcel 2175  {cab 2190

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200

This theorem is referenced by:  rabid  2681  vex  2774  csbco  3102  csbcow  3103  csbnestgf  3145  ifmdc  3611  pwss  3631  snsspw  3804  iunpw  4526  ordon  4533  funcnv3  5335  tfrlem4  6398  tfrlem8  6403  tfrlem9  6404  tfrlemibxssdm  6412  tfr1onlembxssdm  6428  tfrcllembxssdm  6441  ixpm  6816  mapsnen  6902  sbthlem1  7058  1idprl  7702  1idpru  7703  recexprlem1ssl  7745  recexprlem1ssu  7746  recexprlemss1l  7747  recexprlemss1u  7748  txbas  14672