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Theorem frege70 40141
Description: Lemma for frege72 40143. 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 40140 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege70.x . . . 4 𝑋𝑉
32frege68c 40139 . . 3 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴[𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴))))
4 sbcel1v 3842 . . . . 5 ([𝑋 / 𝑥]𝑥𝐴𝑋𝐴)
54biimpri 229 . . . 4 (𝑋𝐴[𝑋 / 𝑥]𝑥𝐴)
6 sbcim1 3828 . . . 4 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → ([𝑋 / 𝑥]𝑥𝐴[𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴)))
7 sbcal 3836 . . . . 5 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴))
8 sbcim1 3828 . . . . . . 7 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦[𝑋 / 𝑥]𝑦𝐴))
9 sbcbr1g 5119 . . . . . . . . 9 (𝑋𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦))
102, 9ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦)
11 csbvarg 4386 . . . . . . . . . 10 (𝑋𝑉𝑋 / 𝑥𝑥 = 𝑋)
122, 11ax-mp 5 . . . . . . . . 9 𝑋 / 𝑥𝑥 = 𝑋
1312breq1i 5069 . . . . . . . 8 (𝑋 / 𝑥𝑥𝑅𝑦𝑋𝑅𝑦)
1410, 13bitri 276 . . . . . . 7 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋𝑅𝑦)
15 sbcg 3850 . . . . . . . 8 (𝑋𝑉 → ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴))
162, 15ax-mp 5 . . . . . . 7 ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴)
178, 14, 163imtr3g 296 . . . . . 6 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → (𝑋𝑅𝑦𝑦𝐴))
1817alimi 1805 . . . . 5 (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
197, 18sylbi 218 . . . 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 207  wal 1528   = wceq 1530  wcel 2106  [wsbc 3775  csb 3886   class class class wbr 5062   hereditary whe 39980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-frege1 39998  ax-frege2 39999  ax-frege8 40017  ax-frege52a 40065  ax-frege58b 40109
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-he 39981
This theorem is referenced by:  frege71  40142
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