Theorem wl-dfrmof 34857

Description: Restricted "at most one" (df-wl-rmo 34854) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)

Assertion

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

Proof of Theorem wl-dfrmof

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-dfralf 34841	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))))
2		nfnfc1 2982	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
3		impexp 453	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
4	3	a1i 11	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))))
5	2, 4	albid 2224	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))))
6	1, 5	bitr4d 284	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)))
7	6	exbidv 1922	. 2 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)))
8		df-wl-rmo 34854	. 2 ⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦))
9		df-mo 2622	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
10	7, 8, 9	3bitr4g 316	1 ⊢ (Ⅎ𝑥𝐴 → (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780   ∈ wcel 2114  ∃*wmo 2620  Ⅎwnfc 2963  ∀wl-ral 34833  ∃*wl-rmo 34835

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-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-mo 2622  df-clel 2895  df-nfc 2965  df-wl-ral 34838  df-wl-rmo 34854

This theorem is referenced by:  wl-dfreuf  34861