Theorem ralab 3367

Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

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 2917	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
3		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	2, 3	elab 3350	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)
5	4	imbi1i 339	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒))
6	5	albii 1747	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒))
7	1, 6	bitri 264	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   ∈ wcel 1990  {cab 2608  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  ralrnmpt2  6775  funcnvuni  7119  kardex  8757  karden  8758  fimaxre3  10970  ptcnp  21425  ptrescn  21442  itg2leub  23501  nmoubi  27627  nmopub  28767  nmfnleub  28784  nmcexi  28885  mblfinlem3  33448  ismblfin  33450  itg2addnc  33464  hbtlem2  37694