Theorem abbi2i 2725

Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 26-May-1993.)

Hypothesis

Ref	Expression
abbi2i.1	⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Assertion

Ref	Expression
abbi2i	⊢ 𝐴 = {𝑥 ∣ 𝜑}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abbi2i

Step	Hyp	Ref	Expression
1		abeq2 2719	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
2		abbi2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	1, 2	mpgbir 1717	1 ⊢ 𝐴 = {𝑥 ∣ 𝜑}

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606

This theorem is referenced by:  abid1  2731  cbvralcsf  3531  cbvreucsf  3533  cbvrabcsf  3534  dfsymdif2  3813  symdif2  3814  dfnul2  3876  dfpr2  4143  dftp2  4178  0iin  4514  epse  5021  fv3  6116  fo1st  7079  fo2nd  7080  xp2  7094  tfrlem3  7361  mapsn  7785  ixpconstg  7803  ixp0x  7822  dfom4  8429  cardnum  8800  alephiso  8804  nnzrab  11282  nn0zrab  11283  qnnen  14781  h2hcau  27220  dfch2  27650  hhcno  28147  hhcnf  28148  pjhmopidm  28426  bdayfo  31074  fobigcup  31177  dfsingles2  31198  dfrecs2  31227  dfrdg4  31228  dfint3  31229  bj-snglinv  32153  compeq  37664