Theorem abeq2i 2345

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 2301	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2222	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220

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 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230

This theorem is referenced by:  rabid  2721  vex  2818  csbco  3151  csbcow  3152  csbnestgf  3194  ifmdc  3669  pwss  3693  snsspw  3873  iunpw  4606  ordon  4613  funcnv3  5423  tfrlem4  6557  tfrlem8  6562  tfrlem9  6563  tfrlemibxssdm  6571  tfr1onlembxssdm  6587  tfrcllembxssdm  6600  ixpm  6978  mapsnen  7066  sbthlem1  7240  1idprl  7921  1idpru  7922  recexprlem1ssl  7964  recexprlem1ssu  7965  recexprlemss1l  7966  recexprlemss1u  7967  txbas  15249