Theorem frege101

Description: Lemma for frege102 . Proposition 101 of Frege1879 p. 72. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege99.z	⊢ 𝑍 ∈ 𝑈
Assertion	frege101	⊢ ( ( 𝑍 = 𝑋 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege99.z	⊢ 𝑍 ∈ 𝑈
2	1	frege100	⊢ ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) )
3		frege48	⊢ ( ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) ) → ( ( 𝑍 = 𝑋 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( 𝑍 = 𝑋 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) → ( 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 → ( 𝑍 𝑅 𝑉 → 𝑋 ( t+ ‘ 𝑅 ) 𝑉 ) ) ) )