Theorem frege72 43278

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.)

Hypotheses

Ref	Expression
frege72.x	⊢ 𝑋 ∈ 𝑈
frege72.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege72	⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

Proof of Theorem frege72

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege72.y	. . . 4 ⊢ 𝑌 ∈ 𝑉
2	1	frege58c 43264	. . 3 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))
3		sbcim1 3830	. . . 4 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧 → [𝑌 / 𝑧]𝑧 ∈ 𝐴))
4		sbcbr2g 5200	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅⦋𝑌 / 𝑧⦌𝑧))
5		csbvarg 4427	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑧⦌𝑧 = 𝑌)
6	5	breq2d 5154	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (𝑋𝑅⦋𝑌 / 𝑧⦌𝑧 ↔ 𝑋𝑅𝑌))
7	4, 6	bitrd 279	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌))
8	1, 7	ax-mp 5	. . . 4 ⊢ ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌)
9		sbcel1v 3844	. . . 4 ⊢ ([𝑌 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴)
10	3, 8, 9	3imtr3g 295	. . 3 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))
11	2, 10	syl 17	. 2 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))
12		frege72.x	. . 3 ⊢ 𝑋 ∈ 𝑈
13	12	frege71 43277	. 2 ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))
14	11, 13	ax-mp 5	1 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   ∈ wcel 2099  [wsbc 3774  ⦋csb 3889   class class class wbr 5142   hereditary whe 43115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-frege1 43133  ax-frege2 43134  ax-frege8 43152  ax-frege52a 43200  ax-frege58b 43244

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-he 43116

This theorem is referenced by:  frege73  43279  frege74  43280