Theorem frege72 43948

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 43934	. . 3 ⊢ (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴))
3		sbcim1 3842	. . . 4 ⊢ ([𝑌 / 𝑧](𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧 → [𝑌 / 𝑧]𝑧 ∈ 𝐴))
4		sbcbr2g 5201	. . . . . 6 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧 ↔ 𝑋𝑅⦋𝑌 / 𝑧⦌𝑧))
5		csbvarg 4434	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑧⦌𝑧 = 𝑌)
6	5	breq2d 5155	. . . . . 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 3856	. . . 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 43947	. 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 2108  [wsbc 3788  ⦋csb 3899   class class class wbr 5143   hereditary whe 43785

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-frege1 43803  ax-frege2 43804  ax-frege8 43822  ax-frege52a 43870  ax-frege58b 43914

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-he 43786

This theorem is referenced by:  frege73  43949  frege74  43950