Theorem rabbidva 2801

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (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 2615	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒))
3		rabbi 2722	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
4	2, 3	sylib 122	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  {crab 2524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-ral 2525  df-rab 2529

This theorem is referenced by:  rabbidv  2802  rabeqbidva  2809  rabbi2dva  3429  rabxfrd  4590  onsucmin  4629  seinxp  4821  fniniseg2  5800  fnniniseg2  5801  f1oresrab  5842  suppval1  6439  mptsuppd  6456  2omap  7269  2omapfi  7271  dfinfre  9230  hashfibclem  11206  minmax  11915  xrminmax  11950  iooinsup  11962  gcdass  12711  lcmass  12782  pcneg  13023  rrgsupp  14411  bdbl  15368  xmetxpbl  15373  lgsquadlem1  15950  lgsquadlem2  15951  2lgslem1a  15961  vtxdfifiun  16292  pw1map  16769