Theorem frege70 44776

Description: Lemma for frege72 44778. 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 44775	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
2		frege70.x	. . . 4 ⊢ 𝑋 ∈ 𝑉
3	2	frege68c 44774	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → [𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))))
4		sbcel1v 3804	. . . . 5 ⊢ ([𝑋 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)
5	4	biimpri 231	. . . 4 ⊢ (𝑋 ∈ 𝐴 → [𝑋 / 𝑥]𝑥 ∈ 𝐴)
6		sbcim1 3792	. . . 4 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → ([𝑋 / 𝑥]𝑥 ∈ 𝐴 → [𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7		sbcal 3798	. . . . 5 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))
8		sbcim1 3792	. . . . . . 7 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦 → [𝑋 / 𝑥]𝑦 ∈ 𝐴))
9		sbcbr1g 5162	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦))
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦)
11		csbvarg 4392	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
12	2, 11	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑥⦌𝑥 = 𝑋
13	12	breq1i 5110	. . . . . . . 8 ⊢ (⦋𝑋 / 𝑥⦌𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
14	10, 13	bitri 278	. . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
15		sbcg 3811	. . . . . . . 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 298	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
18	17	alimi 1844	. . . . 5 ⊢ (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
19	7, 18	sylbi 220	. . . 4 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
20	5, 6, 19	syl56 37	. . 3 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))
21	3, 20	syl6 36	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2145  [wsbc 3739  ⦋csb 3847   class class class wbr 5103   hereditary whe 44615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-frege1 44633  ax-frege2 44634  ax-frege8 44652  ax-frege52a 44700  ax-frege58b 44744

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-he 44616

This theorem is used by:  frege71  44777