Theorem wl-dfrexf 34288

Description: Restricted existential quantification (df-wl-rex 34287) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 25-May-2023.)

Assertion

Ref	Expression
wl-dfrexf	⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))

Proof of Theorem wl-dfrexf

Step	Hyp	Ref	Expression
1		wl-dfralf 34280	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)))
2	1	notbid 310	. 2 ⊢ (Ⅎ𝑥𝐴 → (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)))
3		df-wl-rex 34287	. 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4		exnalimn 1806	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
5	2, 3, 4	3bitr4g 306	1 ⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742   ∈ wcel 2050  Ⅎwnfc 2916  ∀wl-ral 34272  ∃wl-rex 34273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-11 2093  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-clel 2846  df-nfc 2918  df-wl-ral 34277  df-wl-rex 34287

This theorem is referenced by:  wl-dfrexfi  34289  wl-dfreuf  34300