Theorem frege70 44514

Description: Lemma for frege72 44516. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege70.x	⊢ 𝑋 ∈ 𝑉

Assertion

Ref	Expression
frege70	⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem frege70

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffrege69 44513	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
2		frege70.x	. . . 4 ⊢ 𝑋 ∈ 𝑉
3	2	frege68c 44512	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → [𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))))
4		sbcel1v 3811	. . . . 5 ⊢ ([𝑋 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)
5	4	biimpri 230	. . . 4 ⊢ (𝑋 ∈ 𝐴 → [𝑋 / 𝑥]𝑥 ∈ 𝐴)
6		sbcim1 3799	. . . 4 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → ([𝑋 / 𝑥]𝑥 ∈ 𝐴 → [𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7		sbcal 3805	. . . . 5 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))
8		sbcim1 3799	. . . . . . 7 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦 → [𝑋 / 𝑥]𝑦 ∈ 𝐴))
9		sbcbr1g 5159	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦))
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦)
11		csbvarg 4390	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
12	2, 11	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑥⦌𝑥 = 𝑋
13	12	breq1i 5109	. . . . . . . 8 ⊢ (⦋𝑋 / 𝑥⦌𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
14	10, 13	bitri 277	. . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
15		sbcg 3818	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	2, 15	ax-mp 5	. . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
17	8, 14, 16	3imtr3g 297	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
18	17	alimi 1833	. . . . 5 ⊢ (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
19	7, 18	sylbi 219	. . . 4 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
20	5, 6, 19	syl56 36	. . 3 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))
21	3, 20	syl6 35	. 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))))
22	1, 21	ax-mp 5	1 ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   = wceq 1562   ∈ wcel 2144  [wsbc 3746  ⦋csb 3854   class class class wbr 5102   hereditary whe 44353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-frege1 44371  ax-frege2 44372  ax-frege8 44390  ax-frege52a 44438  ax-frege58b 44482

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1075  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-he 44354

This theorem is referenced by:  frege71  44515