Theorem rabbi2dva 3464

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		elin 3220	. . . 4 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2	1	eqabi 2465	. . 3 ⊢ (A ∩ B) = {x ∣ (x ∈ A ∧ x ∈ B)}
3		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ x ∈ B} = {x ∣ (x ∈ A ∧ x ∈ B)}
4	2, 3	eqtr4i 2376	. 2 ⊢ (A ∩ B) = {x ∈ A ∣ x ∈ B}
5		rabbi2dva.1	. . 3 ⊢ ((φ ∧ x ∈ A) → (x ∈ B ↔ ψ))
6	5	rabbidva 2851	. 2 ⊢ (φ → {x ∈ A ∣ x ∈ B} = {x ∈ A ∣ ψ})
7	4, 6	syl5eq 2397	1 ⊢ (φ → (A ∩ B) = {x ∈ A ∣ ψ})

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619   ∩ cin 3209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is used by: (None)