Theorem abbi2i 2311

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
abbiri.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 2305	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
2		abbiri.1	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
3	1, 2	mpgbir 1467	1 ⊢ 𝐴 = {𝑥 ∣ 𝜑}

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192

This theorem is referenced by:  abid2  2317  cbvralcsf  3147  cbvrexcsf  3148  cbvreucsf  3149  cbvrabcsf  3150  symdifxor  3430  dfnul2  3453  dfpr2  3642  dftp2  3672  0iin  3976  pwpwab  4005  epse  4378  fv3  5584  fo1st  6224  fo2nd  6225  xp2  6240  tfrlem3  6378  tfr1onlem3  6405  mapsn  6758  ixpconstg  6775  ixp0x  6794  nnzrab  9367  nn0zrab  9368