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Theorem frege70 43242
Description: Lemma for frege72 43244. 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 43241 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege70.x . . . 4 𝑋𝑉
32frege68c 43240 . . 3 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (𝑅 hereditary 𝐴[𝑋 / 𝑥](𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴))))
4 sbcel1v 3843 . . . . 5 ([𝑋 / 𝑥]𝑥𝐴𝑋𝐴)
54biimpri 227 . . . 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 5198 . . . . . . . . 9 (𝑋𝑉 → ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦))
102, 9ax-mp 5 . . . . . . . 8 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋 / 𝑥𝑥𝑅𝑦)
11 csbvarg 4426 . . . . . . . . . 10 (𝑋𝑉𝑋 / 𝑥𝑥 = 𝑋)
122, 11ax-mp 5 . . . . . . . . 9 𝑋 / 𝑥𝑥 = 𝑋
1312breq1i 5148 . . . . . . . 8 (𝑋 / 𝑥𝑥𝑅𝑦𝑋𝑅𝑦)
1410, 13bitri 275 . . . . . . 7 ([𝑋 / 𝑥]𝑥𝑅𝑦𝑋𝑅𝑦)
15 sbcg 3851 . . . . . . . 8 (𝑋𝑉 → ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴))
162, 15ax-mp 5 . . . . . . 7 ([𝑋 / 𝑥]𝑦𝐴𝑦𝐴)
178, 14, 163imtr3g 295 . . . . . 6 ([𝑋 / 𝑥](𝑥𝑅𝑦𝑦𝐴) → (𝑋𝑅𝑦𝑦𝐴))
1817alimi 1805 . . . . 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 1531   = wceq 1533  wcel 2098  [wsbc 3772  csb 3888   class class class wbr 5141   hereditary whe 43081
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-frege1 43099  ax-frege2 43100  ax-frege8 43118  ax-frege52a 43166  ax-frege58b 43210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-he 43082
This theorem is referenced by:  frege71  43243
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