Theorem wl-dfralf 34854

Description: Restricted universal quantification (df-wl-ral 34851) allows a simplification, if we can assume all appearences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 23-May-2023.)

Assertion

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

Proof of Theorem wl-dfralf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcr 2966	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
2		19.21t 2206	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 → (∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
3	1, 2	syl 17	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
4	3	albidv 1921	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
5		wl-dfralflem 34853	. . . 4 ⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6	5	bicomi 226	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)))
7		df-wl-ral 34851	. . 3 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	4, 6, 7	3bitr4g 316	. 2 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀(𝑥 : 𝐴)𝜑))
9	8	bicomd 225	1 ⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2961  ∀wl-ral 34846

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 2893  df-nfc 2963  df-wl-ral 34851

This theorem is referenced by:  wl-dfralfi  34855  wl-dfrexf  34862  wl-dfrmof  34870