Theorem ralab 3640

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 3053	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		df-clab 2716	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
3		ralab.1	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	3	sbievw 2099	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜓)
5	2, 4	bitri 275	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)
6	5	imbi1i 349	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒))
7	6	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒))
8	1, 7	bitri 275	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  [wsb 2068   ∈ wcel 2114  {cab 2715  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-ral 3053

This theorem is referenced by:  rexab  3642  ralrnmpo  7499  funcnvuni  7876  kardex  9809  karden  9810  fimaxre3  12093  ptcnp  23597  ptrescn  23614  itg2leub  25711  addsuniflem  28007  addbdaylem  28023  mulsuniflem  28155  nmoubi  30858  nmopub  31994  nmfnleub  32011  nmcexi  32112  mblfinlem3  37994  ismblfin  37996  itg2addnc  38009  hbtlem2  43570  oaun3lem1  43820