Theorem rexab 3703

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 3071	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒)
2		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
3	2	ralab 3700	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒 ↔ ∀𝑥(𝜓 → ¬ 𝜒))
4	1, 3	xchbinx 334	. . 3 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥(𝜓 → ¬ 𝜒))
5		imnang 1839	. . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) ↔ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
6	4, 5	xchbinx 334	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
7		df-ex 1777	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
8	6, 7	bitr4i 278	1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1776  {cab 2712  ∀wral 3059  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-ral 3060  df-rex 3069

This theorem is referenced by:  4sqlem12  16990  noinfno  27778  sleadd1  28037  addsuniflem  28049  addsasslem1  28051  addsasslem2  28052  mulsuniflem  28190  addsdilem1  28192  addsdilem2  28193  mulsasslem1  28204  mulsasslem2  28205  renegscl  28445  readdscl  28446  remulscl  28449  mblfinlem3  37646  mblfinlem4  37647  ismblfin  37648  itg2addnclem  37658  itg2addnc  37661  diophrex  42763