Theorem frege112

Description: Identity implies belonging to the R -sequence beginning with self. Proposition 112 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege112.z	⊢ 𝑍 ∈ 𝑉
Assertion	frege112	⊢ ( 𝑍 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		frege112.z	⊢ 𝑍 ∈ 𝑉
2	1	frege105	⊢ ( ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 )
3		frege11	⊢ ( ( ( ¬ 𝑋 ( t+ ‘ 𝑅 ) 𝑍 → 𝑍 = 𝑋 ) → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) → ( 𝑍 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) )
4	2 3	ax-mp	⊢ ( 𝑍 = 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 )