Theorem wl-dfralfi 34842

Description: Restricted universal quantification (df-wl-ral 34838) allows 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-dralfi.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
wl-dfralfi	⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))

Proof of Theorem wl-dfralfi

Step	Hyp	Ref	Expression
1		wl-dralfi.1	. 2 ⊢ Ⅎ𝑥𝐴
2		wl-dfralf 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:   → wi 4   ↔ wb 208  ∀wal 1535   ∈ wcel 2114  Ⅎwnfc 2963  ∀wl-ral 34833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-clel 2895  df-nfc 2965  df-wl-ral 34838

This theorem is referenced by: (None)