Theorem frege70 44036

Description: Lemma for frege72 44038. 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 44035	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
2		frege70.x	. . . 4 ⊢ 𝑋 ∈ 𝑉
3	2	frege68c 44034	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → [𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))))
4		sbcel1v 3802	. . . . 5 ⊢ ([𝑋 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)
5	4	biimpri 228	. . . 4 ⊢ (𝑋 ∈ 𝐴 → [𝑋 / 𝑥]𝑥 ∈ 𝐴)
6		sbcim1 3790	. . . 4 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → ([𝑋 / 𝑥]𝑥 ∈ 𝐴 → [𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7		sbcal 3796	. . . . 5 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))
8		sbcim1 3790	. . . . . . 7 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦 → [𝑋 / 𝑥]𝑦 ∈ 𝐴))
9		sbcbr1g 5146	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦))
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦)
11		csbvarg 4381	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
12	2, 11	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑥⦌𝑥 = 𝑋
13	12	breq1i 5096	. . . . . . . 8 ⊢ (⦋𝑋 / 𝑥⦌𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
14	10, 13	bitri 275	. . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
15		sbcg 3809	. . . . . . . 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 295	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
18	17	alimi 1812	. . . . 5 ⊢ (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
19	7, 18	sylbi 217	. . . 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 206  ∀wal 1539   = wceq 1541   ∈ wcel 2111  [wsbc 3736  ⦋csb 3845   class class class wbr 5089   hereditary whe 43875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-frege1 43893  ax-frege2 43894  ax-frege8 43912  ax-frege52a 43960  ax-frege58b 44004

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-he 43876

This theorem is referenced by:  frege71  44037