Theorem ralab 3676

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 3052	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		df-clab 2714	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
3		ralab.1	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	3	sbievw 2093	. . . . . 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 1819	. 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 1538  [wsb 2064   ∈ wcel 2108  {cab 2713  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-ral 3052

This theorem is referenced by:  rexab  3678  ralrnmpo  7546  funcnvuni  7928  kardex  9908  karden  9909  fimaxre3  12188  ptcnp  23560  ptrescn  23577  itg2leub  25687  addsuniflem  27960  addsbdaylem  27975  mulsuniflem  28104  nmoubi  30753  nmopub  31889  nmfnleub  31906  nmcexi  32007  mblfinlem3  37683  ismblfin  37685  itg2addnc  37698  hbtlem2  43148  oaun3lem1  43398