Theorem rabbi2dva 4179

Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Hypothesis

Ref	Expression
rabbi2dva.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))

Assertion

Ref	Expression
rabbi2dva	⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabbi2dva

Step	Hyp	Ref	Expression
1		dfin5 3914	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
2		rabbi2dva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))
3	2	rabbidva 3422	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝜓})
4	1, 3	eqtrid 2810	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416   ∩ cin 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-rab 3417  df-in 3913

This theorem is referenced by:  fndmdif  7039  bitsshft  16534  sylow3lem2  19699  leordtvallem1  23348  leordtvallem2  23349  ordtt1  23517  xkoccn  23757  txcnmpt  23762  xkopt  23793  ordthmeolem  23939  qustgphaus  24261  itg2monolem1  25890  lhop1  26154  efopn  26804  dirith  27674  pjvec  32029  pjocvec  32030  dfscott3  35493  neibastop3  36854  diarnN  41884