Theorem rexab 3624

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)

Hypothesis

Ref	Expression
ralab.1	⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexab	⊢ (∃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 rexab

Step	Hyp	Ref	Expression
1		dfrex2 3166	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒)
2		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
3	2	ralab 3621	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒 ↔ ∀𝑥(𝜓 → ¬ 𝜒))
4	1, 3	xchbinx 333	. . 3 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥(𝜓 → ¬ 𝜒))
5		imnang 1845	. . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) ↔ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
6	4, 5	xchbinx 333	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
7		df-ex 1784	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
8	6, 7	bitr4i 277	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  {cab 2715  ∀wral 3063  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-ral 3068  df-rex 3069

This theorem is referenced by:  4sqlem12  16585  noinfno  33848  mblfinlem3  35743  mblfinlem4  35744  ismblfin  35745  itg2addnclem  35755  itg2addnc  35758  diophrex  40513