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Theorem frege70 37247
 Description: Lemma for frege72 37249. 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 𝑋𝑉
Assertion
Ref Expression
frege70 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))
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.
StepHypRef Expression
1 dffrege69 37246 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege70.x . . . 4 𝑋𝑉
32frege68c 37245 . . 3 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴[𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴))))
4 sbcel1v 3462 . . . . 5 ([𝑋 / 𝑥]𝑥𝐴𝑋𝐴)
54biimpri 217 . . . 4 (𝑋𝐴[𝑋 / 𝑥]𝑥𝐴)
6 sbcim1 3449 . . . 4 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → ([𝑋 / 𝑥]𝑥𝐴[𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴)))
7 sbcal 3452 . . . . 5 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴))
8 sbcim1 3449 . . . . . . 7 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦[𝑋 / 𝑥]𝑦𝐴))
9 sbcbr1g 4639 . . . . . . . . 9 (𝑋𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦))
102, 9ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦)
11 csbvarg 3955 . . . . . . . . . 10 (𝑋𝑉𝑋 / 𝑥𝑥 = 𝑋)
122, 11ax-mp 5 . . . . . . . . 9 𝑋 / 𝑥𝑥 = 𝑋
1312breq1i 4590 . . . . . . . 8 (𝑋 / 𝑥𝑥𝑅𝑦𝑋𝑅𝑦)
1410, 13bitri 263 . . . . . . 7 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋𝑅𝑦)
15 sbcg 3470 . . . . . . . 8 (𝑋𝑉 → ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴))
162, 15ax-mp 5 . . . . . . 7 ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴)
178, 14, 163imtr3g 283 . . . . . 6 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → (𝑋𝑅𝑦𝑦𝐴))
1817alimi 1730 . . . . 5 (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
197, 18sylbi 206 . . . 4 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
205, 6, 19syl56 35 . . 3 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))
213, 20syl6 34 . 2 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))))
221, 21ax-mp 5 1 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   class class class wbr 4583   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52a 37171  ax-frege58b 37215 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087 This theorem is referenced by:  frege71  37248
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