Theorem frege72 42671

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 42657	. . 3 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))
3		sbcim1 3832	. . . 4 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧 → [𝑌 / 𝑧]𝑧 ∈ 𝐴))
4		sbcbr2g 5205	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅⦋𝑌 / 𝑧⦌𝑧))
5		csbvarg 4430	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑧⦌𝑧 = 𝑌)
6	5	breq2d 5159	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → (𝑋𝑅⦋𝑌 / 𝑧⦌𝑧 ↔ 𝑋𝑅𝑌))
7	4, 6	bitrd 278	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌))
8	1, 7	ax-mp 5	. . . 4 ⊢ ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅𝑌)
9		sbcel1v 3847	. . . 4 ⊢ ([𝑌 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴)
10	3, 8, 9	3imtr3g 294	. . 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 42670	. 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 1539   ∈ wcel 2106  [wsbc 3776  ⦋csb 3892   class class class wbr 5147   hereditary whe 42508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-frege1 42526  ax-frege2 42527  ax-frege8 42545  ax-frege52a 42593  ax-frege58b 42637

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-he 42509

This theorem is referenced by:  frege73  42672  frege74  42673