Theorem abeq2i 2343

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 2299	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2220	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by:  rabid  2719  vex  2816  csbco  3148  csbcow  3149  csbnestgf  3191  ifmdc  3665  pwss  3688  snsspw  3868  iunpw  4601  ordon  4608  funcnv3  5418  tfrlem4  6544  tfrlem8  6549  tfrlem9  6550  tfrlemibxssdm  6558  tfr1onlembxssdm  6574  tfrcllembxssdm  6587  ixpm  6965  mapsnen  7053  sbthlem1  7227  1idprl  7905  1idpru  7906  recexprlem1ssl  7948  recexprlem1ssu  7949  recexprlemss1l  7950  recexprlemss1u  7951  txbas  15123