Theorem ralrab2 3478

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 3023	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2	1	raleqi 3245	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓)
3		ralab2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
4	3	ralab2 3477	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒))
5		impexp 461	. . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ (𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
6	5	albii 1860	. . 3 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
7		df-ral 3019	. . 3 ⊢ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜒) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝜑 → 𝜒)))
8	6, 7	bitr4i 267	. 2 ⊢ (∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) → 𝜒) ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))
9	2, 4, 8	3bitri 286	1 ⊢ (∀𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1594   ∈ wcel 2103  {cab 2710  ∀wral 3014  {crab 3018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rab 3023

This theorem is referenced by:  efgsf  18263  ghmcnp  22040  nmogelb  22642  pntlem3  25418  sstotbnd2  33805