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Theorem frege70 37744
Description: Lemma for frege72 37746. 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 37743 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege70.x . . . 4 𝑋𝑉
32frege68c 37742 . . 3 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴[𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴))))
4 sbcel1v 3481 . . . . 5 ([𝑋 / 𝑥]𝑥𝐴𝑋𝐴)
54biimpri 218 . . . 4 (𝑋𝐴[𝑋 / 𝑥]𝑥𝐴)
6 sbcim1 3468 . . . 4 ([𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → ([𝑋 / 𝑥]𝑥𝐴[𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴)))
7 sbcal 3471 . . . . 5 ([𝑋 / 𝑥]𝑦(𝑥𝑅𝑦𝑦𝐴) ↔ ∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴))
8 sbcim1 3468 . . . . . . 7 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ([𝑋 / 𝑥]𝑥𝑅𝑦[𝑋 / 𝑥]𝑦𝐴))
9 sbcbr1g 4674 . . . . . . . . 9 (𝑋𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦))
102, 9ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦)
11 csbvarg 3980 . . . . . . . . . 10 (𝑋𝑉𝑋 / 𝑥𝑥 = 𝑋)
122, 11ax-mp 5 . . . . . . . . 9 𝑋 / 𝑥𝑥 = 𝑋
1312breq1i 4625 . . . . . . . 8 (𝑋 / 𝑥𝑥𝑅𝑦𝑋𝑅𝑦)
1410, 13bitri 264 . . . . . . 7 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋𝑅𝑦)
15 sbcg 3489 . . . . . . . 8 (𝑋𝑉 → ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴))
162, 15ax-mp 5 . . . . . . 7 ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴)
178, 14, 163imtr3g 284 . . . . . 6 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → (𝑋𝑅𝑦𝑦𝐴))
1817alimi 1736 . . . . 5 (∀𝑦[𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → ∀𝑦(𝑋𝑅𝑦𝑦𝐴))
197, 18sylbi 207 . . . 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 196  wal 1478   = wceq 1480  wcel 1987  [wsbc 3421  csb 3518   class class class wbr 4618   hereditary whe 37583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-frege1 37601  ax-frege2 37602  ax-frege8 37620  ax-frege52a 37668  ax-frege58b 37712
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-he 37584
This theorem is referenced by:  frege71  37745
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