Metamath Proof Explorer


Theorem axprlem3

Description: Lemma for axpr . Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023) (Proof shortened by Matthew House, 18-Sep-2025)

Ref Expression
Assertion axprlem3 𝑧𝑤 ( 𝑤𝑧 ↔ ∃ 𝑠 ( 𝑠𝑝 ∧ if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 axrep4v ( ∀ 𝑠𝑧𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) → ∃ 𝑧𝑤 ( 𝑤𝑧 ↔ ∃ 𝑠 ( 𝑠𝑝 ∧ if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) ) )
2 ifptru ( ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑥 ) )
3 2 biimpd ( ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑥 ) )
4 equeuclr ( 𝑧 = 𝑥 → ( 𝑤 = 𝑥𝑤 = 𝑧 ) )
5 3 4 syl9r ( 𝑧 = 𝑥 → ( ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) )
6 5 alrimdv ( 𝑧 = 𝑥 → ( ∃ 𝑛 𝑛𝑠 → ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) )
7 6 spimevw ( ∃ 𝑛 𝑛𝑠 → ∃ 𝑧𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) )
8 ifpfal ( ¬ ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ↔ 𝑤 = 𝑦 ) )
9 8 biimpd ( ¬ ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑦 ) )
10 equeuclr ( 𝑧 = 𝑦 → ( 𝑤 = 𝑦𝑤 = 𝑧 ) )
11 9 10 syl9r ( 𝑧 = 𝑦 → ( ¬ ∃ 𝑛 𝑛𝑠 → ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) )
12 11 alrimdv ( 𝑧 = 𝑦 → ( ¬ ∃ 𝑛 𝑛𝑠 → ∀ 𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) ) )
13 12 spimevw ( ¬ ∃ 𝑛 𝑛𝑠 → ∃ 𝑧𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 ) )
14 7 13 pm2.61i 𝑧𝑤 ( if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) → 𝑤 = 𝑧 )
15 1 14 mpg 𝑧𝑤 ( 𝑤𝑧 ↔ ∃ 𝑠 ( 𝑠𝑝 ∧ if- ( ∃ 𝑛 𝑛𝑠 , 𝑤 = 𝑥 , 𝑤 = 𝑦 ) ) )