Theorem axprlem3 5318

Description: Lemma for axpr 5321. 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 1922	. . 3 ⊢ Ⅎ𝑧if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)
2	1	axrep4 5184	. 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 2023	. . . 4 ⊢ ∃𝑧 𝑥 = 𝑧
4		ifptru 1076	. . . . . . . . 9 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
5	4	biimpd 232	. . . . . . . 8 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑥))
6		equtrr 2030	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑥 → 𝑤 = 𝑧))
7	5, 6	sylan9r 512	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛 ∈ 𝑠) → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
8	7	alrimiv 1935	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛 ∈ 𝑠) → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
9	8	expcom 417	. . . . 5 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (𝑥 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
10	9	eximdv 1925	. . . 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 2023	. . . 4 ⊢ ∃𝑧 𝑦 = 𝑧
13		ifpfal 1077	. . . . . . . . . 10 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
14	13	biimpd 232	. . . . . . . . 9 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
15	14	adantl 485	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
16		simpl 486	. . . . . . . 8 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → 𝑦 = 𝑧)
17		equtr 2029	. . . . . . . 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 1935	. . . . . 6 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛 ∈ 𝑠) → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
20	19	expcom 417	. . . . 5 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (𝑦 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
21	20	eximdv 1925	. . . 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 185	. 2 ⊢ ∃𝑧∀𝑤(if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)
24	2, 23	mpg 1805	1 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  if-wif 1063  ∀wal 1541  ∃wex 1787   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-rep 5179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ifp 1064  df-tru 1546  df-ex 1788  df-nf 1792

This theorem is referenced by:  axpr  5321