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Theorem frege70 42279
Description: Lemma for frege72 42281. 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 42278 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege70.x . . . 4 𝑋𝑉
32frege68c 42277 . . 3 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴[𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴))))
4 sbcel1v 3815 . . . . 5 ([𝑋 / 𝑥]𝑥𝐴𝑋𝐴)
54biimpri 227 . . . 4 (𝑋𝐴[𝑋 / 𝑥]𝑥𝐴)
6 sbcim1 3800 . . . 4 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → ([𝑋 / 𝑥]𝑥𝐴[𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴)))
7 sbcal 3808 . . . . 5 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴))
8 sbcim1 3800 . . . . . . 7 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦[𝑋 / 𝑥]𝑦𝐴))
9 sbcbr1g 5167 . . . . . . . . 9 (𝑋𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦))
102, 9ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦)
11 csbvarg 4396 . . . . . . . . . 10 (𝑋𝑉𝑋 / 𝑥𝑥 = 𝑋)
122, 11ax-mp 5 . . . . . . . . 9 𝑋 / 𝑥𝑥 = 𝑋
1312breq1i 5117 . . . . . . . 8 (𝑋 / 𝑥𝑥𝑅𝑦𝑋𝑅𝑦)
1410, 13bitri 275 . . . . . . 7 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋𝑅𝑦)
15 sbcg 3823 . . . . . . . 8 (𝑋𝑉 → ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴))
162, 15ax-mp 5 . . . . . . 7 ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴)
178, 14, 163imtr3g 295 . . . . . 6 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → (𝑋𝑅𝑦𝑦𝐴))
1817alimi 1814 . . . . 5 (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
197, 18sylbi 216 . . . 4 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
205, 6, 19syl56 36 . . 3 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))
213, 20syl6 35 . 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 205  wal 1540   = wceq 1542  wcel 2107  [wsbc 3744  csb 3860   class class class wbr 5110   hereditary whe 42118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-frege1 42136  ax-frege2 42137  ax-frege8 42155  ax-frege52a 42203  ax-frege58b 42247
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-he 42119
This theorem is referenced by:  frege71  42280
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