Theorem ralab 3650

Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) Reduce axiom usage. (Revised by GG, 2-Nov-2024.)

Hypothesis

Ref	Expression
ralab.1	⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜓,𝑦

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

Proof of Theorem ralab

Step	Hyp	Ref	Expression
1		df-ral 3046	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		df-clab 2709	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
3		ralab.1	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	3	sbievw 2095	. . . . . 6 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜓)
5	2, 4	bitri 275	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)
6	5	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒))
7		biid 261	. . . 4 ⊢ ((𝜓 → 𝜒) ↔ (𝜓 → 𝜒))
8	6, 7	bitri 275	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒))
9	8	albii 1820	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒))
10	1, 9	bitri 275	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539  [wsb 2066   ∈ wcel 2110  {cab 2708  ∀wral 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-ral 3046

This theorem is referenced by:  rexab  3652  ralrnmpo  7480  funcnvuni  7857  kardex  9779  karden  9780  fimaxre3  12060  ptcnp  23530  ptrescn  23547  itg2leub  25655  addsuniflem  27937  addsbdaylem  27952  mulsuniflem  28081  nmoubi  30742  nmopub  31878  nmfnleub  31895  nmcexi  31996  mblfinlem3  37678  ismblfin  37680  itg2addnc  37693  hbtlem2  43136  oaun3lem1  43386