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Theorem frege71 43924
Description: Lemma for frege72 43925. 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 43923 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴)))
3 frege19 43814 . 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 2106   class class class wbr 5148   hereditary whe 43762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-frege1 43780  ax-frege2 43781  ax-frege8 43799  ax-frege52a 43847  ax-frege58b 43891
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-he 43763
This theorem is referenced by:  frege72  43925
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