Theorem wl-dfrexf 34883

Description: Restricted existential quantification (df-wl-rex 34882) 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 34875	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)))
2	1	notbid 320	. 2 ⊢ (Ⅎ𝑥𝐴 → (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)))
3		df-wl-rex 34882	. 2 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4		exnalimn 1843	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
5	2, 3, 4	3bitr4g 316	1 ⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779   ∈ wcel 2113  Ⅎwnfc 2960  ∀wl-ral 34867  ∃wl-rex 34868

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 1969  ax-7 2014  ax-8 2115  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-clel 2892  df-nfc 2962  df-wl-ral 34872  df-wl-rex 34882

This theorem is referenced by:  wl-dfrexfi  34884  wl-dfreuf  34895