Theorem rabbidv 2852

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)

Hypothesis

Ref	Expression
rabbidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem rabbidv

Step	Hyp	Ref	Expression
1		rabbidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	adantr 451	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	rabbidva 2851	1 ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {crab 2619

This theorem was proved from 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 theorem 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 referenced by:  rabeqbidv  2855  nmembers1  6272