Theorem ralab 3648

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 3049	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		df-clab 2712	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
3		ralab.1	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	3	sbievw 2098	. . . . . 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 2067   ∈ wcel 2113  {cab 2711  ∀wral 3048

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 2068  df-clab 2712  df-ral 3049

This theorem is referenced by:  rexab  3650  ralrnmpo  7494  funcnvuni  7871  kardex  9798  karden  9799  fimaxre3  12079  ptcnp  23557  ptrescn  23574  itg2leub  25682  addsuniflem  27964  addsbdaylem  27979  mulsuniflem  28108  nmoubi  30773  nmopub  31909  nmfnleub  31926  nmcexi  32027  mblfinlem3  37772  ismblfin  37774  itg2addnc  37787  hbtlem2  43281  oaun3lem1  43531