Theorem birabrdv 1648

Description: Deduction from wff to restricted class abstraction.

Hypothesis

Ref	Expression
birabrdv.1	⊢ (φ → (x ∈ B → (x ∈ A ↔ χ)))

Assertion

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

Distinct variable group(s):   φ,x   x,A   x,B

Proof of Theorem birabrdv

Step	Hyp	Ref	Expression
1		birabrdv.1	. . . . 5 ⊢ (φ → (x ∈ B → (x ∈ A ↔ χ)))
2	1	pm5.32d 491	. . . 4 ⊢ (φ → ((x ∈ B ∧ x ∈ A) ↔ (x ∈ B ∧ χ)))
3		elin 1635	. . . 4 ⊢ (x ∈ (B ∩ A) ↔ (x ∈ B ∧ x ∈ A))
4	2, 3	syl5bb 410	. . 3 ⊢ (φ → (x ∈ (B ∩ A) ↔ (x ∈ B ∧ χ)))
5	4	biabrdv 1184	. 2 ⊢ (φ → (B ∩ A) = {x∣(x ∈ B ∧ χ)})
6		df-rab 1208	. . 3 ⊢ {x ∈ B∣χ} = {x∣(x ∈ B ∧ χ)}
7	6	cleqcomi 1105	. 2 ⊢ {x∣(x ∈ B ∧ χ)} = {x ∈ B∣χ}
8	5, 7	syl6eq 1140	1 ⊢ (φ → (B ∩ A) = {x ∈ B∣χ})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204   ∩ cin 1486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-in 1491

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