Metamath Proof Explorer


Theorem frege70

Description: Lemma for frege72 . Proposition 70 of Frege1879 p. 58. (Contributed by RP, 28-Mar-2020) (Revised by RP, 3-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege70.x 𝑋𝑉
Assertion frege70 ( 𝑅 hereditary 𝐴 → ( 𝑋𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 frege70.x 𝑋𝑉
2 dffrege69 ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) ↔ 𝑅 hereditary 𝐴 )
3 1 frege68c ( ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) → ( 𝑅 hereditary 𝐴[ 𝑋 / 𝑥 ] ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) ) )
4 sbcel1v ( [ 𝑋 / 𝑥 ] 𝑥𝐴𝑋𝐴 )
5 4 biimpri ( 𝑋𝐴[ 𝑋 / 𝑥 ] 𝑥𝐴 )
6 sbcim1 ( [ 𝑋 / 𝑥 ] ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) → ( [ 𝑋 / 𝑥 ] 𝑥𝐴[ 𝑋 / 𝑥 ]𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) )
7 sbcal ( [ 𝑋 / 𝑥 ]𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ↔ ∀ 𝑦 [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦𝑦𝐴 ) )
8 sbcim1 ( [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦𝑦𝐴 ) → ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦[ 𝑋 / 𝑥 ] 𝑦𝐴 ) )
9 sbcbr1g ( 𝑋𝑉 → ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 𝑋 / 𝑥 𝑥 𝑅 𝑦 ) )
10 1 9 ax-mp ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 𝑋 / 𝑥 𝑥 𝑅 𝑦 )
11 csbvarg ( 𝑋𝑉 𝑋 / 𝑥 𝑥 = 𝑋 )
12 1 11 ax-mp 𝑋 / 𝑥 𝑥 = 𝑋
13 12 breq1i ( 𝑋 / 𝑥 𝑥 𝑅 𝑦𝑋 𝑅 𝑦 )
14 10 13 bitri ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦𝑋 𝑅 𝑦 )
15 sbcg ( 𝑋𝑉 → ( [ 𝑋 / 𝑥 ] 𝑦𝐴𝑦𝐴 ) )
16 1 15 ax-mp ( [ 𝑋 / 𝑥 ] 𝑦𝐴𝑦𝐴 )
17 8 14 16 3imtr3g ( [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦𝑦𝐴 ) → ( 𝑋 𝑅 𝑦𝑦𝐴 ) )
18 17 alimi ( ∀ 𝑦 [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦𝑦𝐴 ) → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) )
19 7 18 sylbi ( [ 𝑋 / 𝑥 ]𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) )
20 5 6 19 syl56 ( [ 𝑋 / 𝑥 ] ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) → ( 𝑋𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) ) )
21 3 20 syl6 ( ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦𝑦𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑋𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) ) ) )
22 2 21 ax-mp ( 𝑅 hereditary 𝐴 → ( 𝑋𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦𝑦𝐴 ) ) )