Theorem abeq2i 2281

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 2237	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2158	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	2, 3	bitri 183	1 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Syntax hints:   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {cab 2156

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  rabid  2645  vex  2733  csbco  3059  csbcow  3060  csbnestgf  3101  ifmdc  3565  pwss  3582  snsspw  3751  iunpw  4465  ordon  4470  funcnv3  5260  tfrlem4  6292  tfrlem8  6297  tfrlem9  6298  tfrlemibxssdm  6306  tfr1onlembxssdm  6322  tfrcllembxssdm  6335  ixpm  6708  mapsnen  6789  sbthlem1  6934  1idprl  7552  1idpru  7553  recexprlem1ssl  7595  recexprlem1ssu  7596  recexprlemss1l  7597  recexprlemss1u  7598  txbas  13052