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 3077	. 2 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒))
2		df-clab 2739	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
3		ralab.1	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
4	3	sbievw 2130	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜓)
5	2, 4	bitri 278	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓)
6	5	imbi1i 352	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ (𝜓 → 𝜒))
7	6	albii 1851	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} → 𝜒) ↔ ∀𝑥(𝜓 → 𝜒))
8	1, 7	bitri 278	1 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  [wsb 2098   ∈ wcel 2144  {cab 2738  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-ral 3077

This theorem is used by:  rexab  3652  ralrnmpo  7560  funcnvuni  7945  kardexOLD  9962  kardenOLD  9964  fimaxre3  12267  ptcnp  23947  ptrescn  23964  itg2leub  26061  addsuniflem  28393  addbdaylem  28409  mulsuniflem  28541  nmoubi  31380  nmopub  32516  nmfnleub  32533  nmcexi  32634  mblfinlem3  38671  ismblfin  38673  itg2addnc  38686  hbtlem2  44220  oaun3lem1  44470