Theorem rabbi2dva 4171

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 3907	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
2		rabbi2dva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))
3	2	rabbidva 3418	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝜓})
4	1, 3	eqtrid 2807	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-rab 3413  df-in 3906

This theorem is used by:  fndmdif  7034  bitsshft  16565  sylow3lem2  19755  leordtvallem1  23435  leordtvallem2  23436  ordtt1  23604  xkoccn  23845  txcnmpt  23850  xkopt  23881  ordthmeolem  24027  qustgphaus  24349  itg2monolem1  25978  lhop1  26241  efopn  26895  dirith  27765  pjvec  32177  pjocvec  32178  dfscott3  35626  neibastop3  36981  diarnN  42002