Theorem frege70 41430

Description: Lemma for frege72 41432. 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 41429	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
2		frege70.x	. . . 4 ⊢ 𝑋 ∈ 𝑉
3	2	frege68c 41428	. . 3 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → [𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))))
4		sbcel1v 3783	. . . . 5 ⊢ ([𝑋 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)
5	4	biimpri 227	. . . 4 ⊢ (𝑋 ∈ 𝐴 → [𝑋 / 𝑥]𝑥 ∈ 𝐴)
6		sbcim1 3767	. . . 4 ⊢ ([𝑋 / 𝑥](𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → ([𝑋 / 𝑥]𝑥 ∈ 𝐴 → [𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
7		sbcal 3776	. . . . 5 ⊢ ([𝑋 / 𝑥]∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴))
8		sbcim1 3767	. . . . . . 7 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦 → [𝑋 / 𝑥]𝑦 ∈ 𝐴))
9		sbcbr1g 5127	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦))
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ ⦋𝑋 / 𝑥⦌𝑥𝑅𝑦)
11		csbvarg 4362	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌𝑥 = 𝑋)
12	2, 11	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑥⦌𝑥 = 𝑋
13	12	breq1i 5077	. . . . . . . 8 ⊢ (⦋𝑋 / 𝑥⦌𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
14	10, 13	bitri 274	. . . . . . 7 ⊢ ([𝑋 / 𝑥]𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦)
15		sbcg 3791	. . . . . . . 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 294	. . . . . 6 ⊢ ([𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
18	17	alimi 1815	. . . . 5 ⊢ (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦 → 𝑦 ∈ 𝐴) → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴))
19	7, 18	sylbi 216	. . . 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 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  [wsbc 3711  ⦋csb 3828   class class class wbr 5070   hereditary whe 41269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege52a 41354  ax-frege58b 41398

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-he 41270

This theorem is referenced by:  frege71  41431