Theorem frege72

Description: If property A is hereditary in the R -sequence, if x has property A , and if y is a result of an application of the procedure R to x , then y has property A . Proposition 72 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege72.x	⊢ 𝑋 ∈ 𝑈
	frege72.y	⊢ 𝑌 ∈ 𝑉
Assertion	frege72	⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege72.x	⊢ 𝑋 ∈ 𝑈
2		frege72.y	⊢ 𝑌 ∈ 𝑉
3	2	frege58c	⊢ ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → [ 𝑌 / 𝑧 ] ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) )
4		sbcim1	⊢ ( [ 𝑌 / 𝑧 ] ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( [ 𝑌 / 𝑧 ] 𝑋 𝑅 𝑧 → [ 𝑌 / 𝑧 ] 𝑧 ∈ 𝐴 ) )
5		sbcbr2g	⊢ ( 𝑌 ∈ 𝑉 → ( [ 𝑌 / 𝑧 ] 𝑋 𝑅 𝑧 ↔ 𝑋 𝑅 ⦋ 𝑌 / 𝑧 ⦌ 𝑧 ) )
6		csbvarg	⊢ ( 𝑌 ∈ 𝑉 → ⦋ 𝑌 / 𝑧 ⦌ 𝑧 = 𝑌 )
7	6	breq2d	⊢ ( 𝑌 ∈ 𝑉 → ( 𝑋 𝑅 ⦋ 𝑌 / 𝑧 ⦌ 𝑧 ↔ 𝑋 𝑅 𝑌 ) )
8	5 7	bitrd	⊢ ( 𝑌 ∈ 𝑉 → ( [ 𝑌 / 𝑧 ] 𝑋 𝑅 𝑧 ↔ 𝑋 𝑅 𝑌 ) )
9	2 8	ax-mp	⊢ ( [ 𝑌 / 𝑧 ] 𝑋 𝑅 𝑧 ↔ 𝑋 𝑅 𝑌 )
10		sbcel1v	⊢ ( [ 𝑌 / 𝑧 ] 𝑧 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴 )
11	4 9 10	3imtr3g	⊢ ( [ 𝑌 / 𝑧 ] ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) )
12	3 11	syl	⊢ ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) )
13	1	frege71	⊢ ( ( ∀ 𝑧 ( 𝑋 𝑅 𝑧 → 𝑧 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )
14	12 13	ax-mp	⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ( 𝑋 𝑅 𝑌 → 𝑌 ∈ 𝐴 ) ) )