Theorem ralrab2 3661

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 3415	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2	1	raleqi 3318	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓)
3		ralab2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
4	3	ralab2 3660	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒))
5		impexp 454	. . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ (𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
6	5	albii 1839	. . 3 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
7		df-ral 3077	. . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
8	6, 7	bitr4i 280	. 2 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
9	2, 4, 8	3bitri 299	1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   ∈ wcel 2142  {cab 2740  ∀wral 3076  {crab 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-ral 3077  df-rex 3087  df-rab 3415

This theorem is referenced by:  efgsf  19769  ghmcnp  24175  nmogelb  24776  pntlem3  27673  sstotbnd2  38273