Theorem rexab 3659

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 GG, 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 3092	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒)
2		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
3	2	ralab 3657	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒 ↔ ∀𝑥(𝜓 → ¬ 𝜒))
4	1, 3	xchbinx 337	. . 3 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥(𝜓 → ¬ 𝜒))
5		imnang 1872	. . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) ↔ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
6	4, 5	xchbinx 337	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
7		df-ex 1810	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
8	6, 7	bitr4i 281	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃wex 1809  {cab 2741  ∀wral 3079  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-ral 3080  df-rex 3090

This theorem is referenced by:  4sqlem12  17017  noinfno  27860  leadds1  28160  addsuniflem  28172  addsasslem1  28174  addsasslem2  28175  mulsuniflem  28320  addsdilem1  28322  addsdilem2  28323  mulsasslem1  28334  mulsasslem2  28335  elreno2  28666  renegscl  28669  readdscl  28670  remulscl  28673  mblfinlem3  38288  mblfinlem4  38289  ismblfin  38290  itg2addnclem  38300  itg2addnc  38303  diophrex  43486