Theorem rabbidva 2851

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Hypothesis

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

Assertion

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

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem rabbidva

Step	Hyp	Ref	Expression
1		rabbidva.1	. . 3 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
2	1	ralrimiva 2698	. 2 ⊢ (φ → ∀x ∈ A (ψ ↔ χ))
3		rabbi 2790	. 2 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
4	2, 3	sylib 188	1 ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  {crab 2619

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2620  df-rab 2624

This theorem is used by:  rabbidv  2852  rabeqbidva  2856  rabbi2dva  3464