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Theorem frege71 40415
Description: Lemma for frege72 40416. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege71.x 𝑋𝑉
Assertion
Ref Expression
frege71 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋
Allowed substitution hints:   𝑉(𝑧)   𝑌(𝑧)

Proof of Theorem frege71
StepHypRef Expression
1 frege71.x . . 3 𝑋𝑉
21frege70 40414 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴)))
3 frege19 40305 . 2 ((𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴))) → ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))))
42, 3ax-mp 5 1 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wcel 2114   class class class wbr 5042   hereditary whe 40253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-frege1 40271  ax-frege2 40272  ax-frege8 40290  ax-frege52a 40338  ax-frege58b 40382
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-he 40254
This theorem is referenced by:  frege72  40416
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