Theorem rabbi2dva 4154

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 3891	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵}
2		rabbi2dva.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))
3	2	rabbidva 3397	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝜓})
4	1, 3	eqtrid 2786	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3391   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-rab 3392  df-in 3890

This theorem is referenced by:  fndmdif  6983  bitsshft  16435  sylow3lem2  19594  leordtvallem1  23193  leordtvallem2  23194  ordtt1  23362  xkoccn  23602  txcnmpt  23607  xkopt  23638  ordthmeolem  23784  qustgphaus  24106  itg2monolem1  25735  lhop1  25999  efopn  26640  dirith  27510  pjvec  31785  pjocvec  31786  neibastop3  36590  diarnN  41621