Theorem abeq2i 2342

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 2298	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2219	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {cab 2217

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227

This theorem is referenced by:  rabid  2709  vex  2805  csbco  3137  csbcow  3138  csbnestgf  3180  ifmdc  3648  pwss  3668  snsspw  3847  iunpw  4577  ordon  4584  funcnv3  5392  tfrlem4  6478  tfrlem8  6483  tfrlem9  6484  tfrlemibxssdm  6492  tfr1onlembxssdm  6508  tfrcllembxssdm  6521  ixpm  6898  mapsnen  6985  sbthlem1  7155  1idprl  7809  1idpru  7810  recexprlem1ssl  7852  recexprlem1ssu  7853  recexprlemss1l  7854  recexprlemss1u  7855  txbas  14981