Theorem rexab 3683

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 3064	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒)
2		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
3	2	ralab 3681	. . . 4 ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑} ¬ 𝜒 ↔ ∀𝑥(𝜓 → ¬ 𝜒))
4	1, 3	xchbinx 334	. . 3 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥(𝜓 → ¬ 𝜒))
5		imnang 1842	. . 3 ⊢ (∀𝑥(𝜓 → ¬ 𝜒) ↔ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
6	4, 5	xchbinx 334	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ¬ ∀𝑥 ¬ (𝜓 ∧ 𝜒))
7		df-ex 1780	. 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 1538  ∃wex 1779  {cab 2714  ∀wral 3052  ∃wrex 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-ral 3053  df-rex 3062

This theorem is referenced by:  4sqlem12  16981  noinfno  27687  sleadd1  27953  addsuniflem  27965  addsasslem1  27967  addsasslem2  27968  mulsuniflem  28109  addsdilem1  28111  addsdilem2  28112  mulsasslem1  28123  mulsasslem2  28124  renegscl  28406  readdscl  28407  remulscl  28410  mblfinlem3  37688  mblfinlem4  37689  ismblfin  37690  itg2addnclem  37700  itg2addnc  37703  diophrex  42765