Theorem birabdv 1343

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule).

Hypothesis

Ref	Expression
birabdv.1	⊢ (φ → (x ∈ A → (ψ ↔ χ)))

Assertion

Ref	Expression
birabdv	⊢ (φ → {x ∈ A∣ψ} = {x ∈ A∣χ})

Distinct variable group(s):   φ,x

Proof of Theorem birabdv

Step	Hyp	Ref	Expression
1		birabdv.1	. . . 4 ⊢ (φ → (x ∈ A → (ψ ↔ χ)))
2	1	pm5.32d 491	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
3	2	biabdv 1183	. 2 ⊢ (φ → {x∣(x ∈ A ∧ ψ)} = {x∣(x ∈ A ∧ χ)})
4		df-rab 1208	. 2 ⊢ {x ∈ A∣ψ} = {x∣(x ∈ A ∧ ψ)}
5		df-rab 1208	. 2 ⊢ {x ∈ A∣χ} = {x∣(x ∈ A ∧ χ)}
6	3, 4, 5	3eqtr4g 1147	1 ⊢ (φ → {x ∈ A∣ψ} = {x ∈ A∣χ})

This theorem is referenced by:  birabsdv 1344  onsucmin 2323

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq DSPAN CLASS=r STYLE="color:#D8A800">1091   ∈ wcel 1092  {crab 1204

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

metamath.org