Theorem axprlem3 5348

Description: Lemma for axpr 5351. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.)

Assertion

Ref	Expression
axprlem3	⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤   𝑧,𝑛,𝑤   𝑧,𝑠,𝑝,𝑤

Proof of Theorem axprlem3

Step	Hyp	Ref	Expression
1		nfv 1917	. . 3 ⊢ Ⅎ𝑧if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)
2	1	axrep4 5214	. 2 ⊢ (∀𝑠∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧) → ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
3		ax6evr 2018	. . . 4 ⊢ ∃𝑧 𝑥 = 𝑧
4		ifptru 1073	. . . . . . . . 9 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
5	4	biimpd 228	. . . . . . . 8 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑥))
6		equtrr 2025	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑥 → 𝑤 = 𝑧))
7	5, 6	sylan9r 509	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛 ∈ 𝑠) → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
8	7	alrimiv 1930	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛 ∈ 𝑠) → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
9	8	expcom 414	. . . . 5 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (𝑥 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
10	9	eximdv 1920	. . . 4 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (∃𝑧 𝑥 = 𝑧 → ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
11	3, 10	mpi 20	. . 3 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
12		ax6evr 2018	. . . 4 ⊢ ∃𝑧 𝑦 = 𝑧
13		ifpfal 1074	. . . . . . . . . 10 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
14	13	biimpd 228	. . . . . . . . 9 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
15	14	adantl 482	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
16		simpl 483	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → 𝑦 = 𝑧)
17		equtr 2024	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑦 = 𝑧 → 𝑤 = 𝑧))
18	15, 16, 17	syl6ci 71	. . . . . . 7 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
19	18	alrimiv 1930	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
20	19	expcom 414	. . . . 5 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (𝑦 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
21	20	eximdv 1920	. . . 4 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
22	12, 21	mpi 20	. . 3 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
23	11, 22	pm2.61i 182	. 2 ⊢ ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)
24	2, 23	mpg 1800	1 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  if-wif 1060  ∀wal 1537  ∃wex 1782   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-rep 5209

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  axpr  5351