Theorem rabbi2dva 3964

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 3723	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
2		rabbi2dva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))
3	2	rabbidva 3328	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝜓})
4	1, 3	syl5eq 2806	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054   ∩ cin 3714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rab 3059  df-in 3722

This theorem is referenced by:  fndmdif  6484  bitsshft  15399  sylow3lem2  18243  leordtvallem1  21216  leordtvallem2  21217  ordtt1  21385  xkoccn  21624  txcnmpt  21629  xkopt  21660  ordthmeolem  21806  qustgphaus  22127  itg2monolem1  23716  lhop1  23976  efopn  24603  dirith  25417  pjvec  28864  pjocvec  28865  neibastop3  32663  diarnN  36920