Theorem ralrab2 2802

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralrab2	⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦

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

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2384	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2	1	raleqi 2588	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓)
3		ralab2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
4	3	ralab2 2801	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒))
5		impexp 261	. . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ (𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
6	5	albii 1414	. . 3 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
7		df-ral 2380	. . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
8	6, 7	bitr4i 186	. 2 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
9	2, 4, 8	3bitri 205	1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1297   ∈ wcel 1448  {cab 2086  ∀wral 2375  {crab 2379

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384

This theorem is referenced by: (None)