Theorem abeq2i 2268

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 2224	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2145	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 183	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 104   = wceq 1335   ∈ wcel 2128  {cab 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153

This theorem is referenced by:  rabid  2632  vex  2715  csbco  3041  csbcow  3042  csbnestgf  3083  ifmdc  3542  pwss  3559  snsspw  3727  iunpw  4440  ordon  4445  funcnv3  5232  tfrlem4  6260  tfrlem8  6265  tfrlem9  6266  tfrlemibxssdm  6274  tfr1onlembxssdm  6290  tfrcllembxssdm  6303  ixpm  6675  mapsnen  6756  sbthlem1  6901  1idprl  7510  1idpru  7511  recexprlem1ssl  7553  recexprlem1ssu  7554  recexprlemss1l  7555  recexprlemss1u  7556  txbas  12658