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Theorem frege71 43242
Description: Lemma for frege72 43243. 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 43241 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴)))
3 frege19 43132 . 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 1531  wcel 2098   class class class wbr 5141   hereditary whe 43080
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 43098  ax-frege2 43099  ax-frege8 43117  ax-frege52a 43165  ax-frege58b 43209
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 43081
This theorem is referenced by:  frege72  43243
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