Theorem frege72 43918

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 43904	. . 3 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))
3		sbcim1 3796	. . . 4 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧 → [𝑌 / 𝑧]𝑧 ∈ 𝐴))
4		sbcbr2g 5150	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅⦋𝑌 / 𝑧⦌𝑧))
5		csbvarg 4385	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑧⦌𝑧 = 𝑌)
6	5	breq2d 5104	. . . . . 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 3808	. . . 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 43917	. 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 206  ∀wal 1538   ∈ wcel 2109  [wsbc 3742  ⦋csb 3851   class class class wbr 5092   hereditary whe 43755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-frege1 43773  ax-frege2 43774  ax-frege8 43792  ax-frege52a 43840  ax-frege58b 43884

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-he 43756

This theorem is referenced by:  frege73  43919  frege74  43920