Theorem frege117

Description: Lemma for frege118 . Proposition 117 of Frege1879 p. 78. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege116.x	⊢ 𝑋 ∈ 𝑈
Assertion	frege117	⊢ ( ( ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege116.x	⊢ 𝑋 ∈ 𝑈
2	1	frege116	⊢ ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) )
3		frege9	⊢ ( ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) → ( ( ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) )