Theorem wl-dfrexfi 34842

Description: Restricted universal quantification (df-wl-rex 34840) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)

Hypothesis

Ref	Expression
wl-drexfi.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
wl-dfrexfi	⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Proof of Theorem wl-dfrexfi

Step	Hyp	Ref	Expression
1		wl-drexfi.1	. 2 ⊢ Ⅎ𝑥𝐴
2		wl-dfrexf 34841	. 2 ⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  Ⅎwnfc 2961  ∃wl-rex 34826

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-11 2157  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-clel 2893  df-nfc 2963  df-wl-ral 34830  df-wl-rex 34840

This theorem is referenced by: (None)