Theorem axprglem 5372

Description: Lemma for axprg 5373. (Contributed by GG, 11-Mar-2026.)

Assertion

Ref	Expression
axprglem	⊢ (𝑥 = 𝐴 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))

Distinct variable groups:   𝑥,𝐴,𝑧,𝑤   𝑥,𝐵,𝑧,𝑤

Proof of Theorem axprglem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqsetv-clel 2819	. . . 4 ⊢ (∃𝑦 𝑦 = 𝐵 ↔ ∃𝑤 𝑤 = 𝐵)
2		ax-pr 5369	. . . . . 6 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
3		eqtr3 2762	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐵 ∧ 𝑦 = 𝐵) → 𝑤 = 𝑦)
4	3	expcom 414	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑤 = 𝐵 → 𝑤 = 𝑦))
5	4	orim2d 974	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
6	5	imim1d 82	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
7	6	alimdv 1923	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
8	7	eximdv 1924	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
9	2, 8	mpi 20	. . . . 5 ⊢ (𝑦 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
10	9	exlimiv 1937	. . . 4 ⊢ (∃𝑦 𝑦 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
11	1, 10	sylbir 236	. . 3 ⊢ (∃𝑤 𝑤 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
12		ax-pr 5369	. . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧)
13		alnex 1788	. . . . . 6 ⊢ (∀𝑤 ¬ 𝑤 = 𝐵 ↔ ¬ ∃𝑤 𝑤 = 𝐵)
14		orel2 896	. . . . . . . 8 ⊢ (¬ 𝑤 = 𝐵 → ((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 = 𝑥))
15		pm2.67-2 897	. . . . . . . 8 ⊢ (((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧) → (𝑤 = 𝑥 → 𝑤 ∈ 𝑧))
16	14, 15	syl9 77	. . . . . . 7 ⊢ (¬ 𝑤 = 𝐵 → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
17	16	al2imi 1822	. . . . . 6 ⊢ (∀𝑤 ¬ 𝑤 = 𝐵 → (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
18	13, 17	sylbir 236	. . . . 5 ⊢ (¬ ∃𝑤 𝑤 = 𝐵 → (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
19	18	eximdv 1924	. . . 4 ⊢ (¬ ∃𝑤 𝑤 = 𝐵 → (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑥) → 𝑤 ∈ 𝑧) → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
20	12, 19	mpi 20	. . 3 ⊢ (¬ ∃𝑤 𝑤 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
21	11, 20	pm2.61i 183	. 2 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)
22		eqtr3 2762	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑤 = 𝑥)
23	22	expcom 414	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑤 = 𝐴 → 𝑤 = 𝑥))
24	23	orim1d 973	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → (𝑤 = 𝑥 ∨ 𝑤 = 𝐵)))
25	24	imim1d 82	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧) → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
26	25	alimdv 1923	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧) → ∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
27	26	eximdv 1924	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)))
28	21, 27	mpi 20	1 ⊢ (𝑥 = 𝐴 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 853  ∀wal 1545   = wceq 1547  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815

This theorem is referenced by:  axprg  5373